Index

Abs is suitable for both symbolic and numerical manipulation.
If z is an exact numerical expression, the numerical estimation of z is used to decide about the sign.
For special cases of z, some simplifications are performed.
Abs has the attribute Listable.

Examples

Abs(-123)123

Abs(1.2 + 3.4*I)3.60555

Abs(3+4*I)5

Abs uses the numerical estimation of an expression to decide about the sign of the result.

Abs(Sin(4))-Sin(4)

-Sin(4) // Calculate0.756802

Simplifications are performed if possible.

Abs(-2*x)2*Abs(x)

Abs(x^3)Abs(x)^3

Accuracyfunction

Syntax

Accuracy(z)

gives the number of precise decimal digits right of the decimal point of the number z.

Details

In case of integers and rational numbers, Accuracy gives Infinity.
In case of real numbers, Accuracy is equal to Precision(z) - Log(10, Abs(z)) - 1.
In case of complex numbers, Accuracy gives lowest accuracy of real part and imaginary part.
In case of an expression, Accuracy gives lowest accuracy of all elements.

Examples

Determine the number's accuracy:

Accuracy(123)Infinity

Accuracy(3/7)Infinity

Accuracy(3.14)14.4577

Accuracy(2.0e24)-9.34644

Accuracy(Overflow)-Infinity

Accuracy(1234567890123456.7890)4.0

Accuracy(4.0 + 3*I)14.3525

Allconstant

Syntax

All

can be used to specifies all elements.

Details

All can usually be used in Part[…] to pick all element from a particular level.

Examples

Part( {{a1,a2}, {b1,b2}, {c1,c2}} , All, 1 ){a1, b1, c1}

{"Hello", "World!"}[All, 1]{"H", "W"}

Alternatives|function

Syntax

Alternatives(pat₁, pat₂, … )pat₁ | pat₂ | …

is a pattern object that represents any pattern pat_i during a pattern matching.

Details

Alternatives can be understand as an OR operation for pattern objects, similar to Or|| for logical and binary expressions.

Examples

Matching( b, a|b )True

Alternatives can be used for replacements:

Replace( {a, b, c, a, b, c} , a|b -> 0 ){0, 0, c, 0, 0, c}

A definition of a function which excepts only integer and rational numbers.

f( x?Rational | x?Integer ) := x^2; { f(2), f(1.0), f(1/2) }{4, f(1.0), 1/4}

And&&function

Syntax

And(bool₁, bool₂, … )bool₁ && bool₂ && …

gives the logical AND of bool_i.

And(int₁, int₂, … )int₁ && int₂ && …

gives the binary AND of the integers int_i.

And(str₁, str₂, … )str₁ && str₂ && …

gives the binary AND of the strings str_i.

Details

The arguments of And are evaluated step by step.
And gives False immediately if any argument is False, and True if all arguments are True.
And() is defined as True.
And(exp) is simplified to exp.
Integers are handled in their two's complement form.
Strings are converted into binary form and And is performed bitwise.

Examples

True && FalseFalse

12 && 74

'BINARY' && 'abc''@@B'

And stops to evaluate its arguments if False occurs. In the following case the second Print is not evaluated.

(Print(1);True) && False && (Print(2);True)1False

Expressions which are evaluated into True will be erased.

2==2 && a==5 && b==10 && Truea == 5 && b == 10

Any?function

Syntax

Any?

is a pattern object that matches to any object.

Any(head)?head

matches to any object having the head head.

Details

Any is typical used in assignments and rules.

Examples

Matching( f(123), Any )True

Matching( f(123), Any(f) )True

Any can used in rules for replacements.

Replace( { 1, 1.0, 1/2, 4 } , ?Integer -> x ){x, 1.0, 1/2, x}

Any can used in an assignment as a pattern for an argument.

f(?Real) = "real"; { f(1), f(1.0), f(2^0.5) }{f(1), "real", "real"}

Possible Issues

In functions like Plus+ and Times*, Any does not differ from other symbols and is transformed.

? + ? Matching( a + b, ? + ? ) Matching( a + b, x? + y? )2*Any False True

AnyNullSequence???function

Syntax

AnyNullSequence???

is a pattern object that matches to a sequence of any objects, even an empty sequence of no object.

AnyNullSequence(head)???head

matches to a sequence of objects all having the head head, or an empty sequence.

Details

AnyNullSequence is typical used in assignments and rules.

Examples

Matching( f(), f(AnyNullSequence) )True

Matching( f(x, y), f(AnyNullSequence(Symbol)) )True

Matching( f(), f(AnyNullSequence(Symbol)) )True

AnyNullSequence can used in an assignment as a pattern for a sequence of arguments.

f(???Real) = "reals or nothing"; { f(), f(1.0), f(1.0, 2.0), f(1) }{"reals or nothing", "reals or nothing", "reals or nothing", f(1)}

AnySequence??function

Syntax

AnySequence??

is a pattern object that matches to a sequence of any objects.

AnySequence(head)??head

matches to a sequence of objects all having the head head.

Details

AnySequence is typical used in assignments and rules.

Examples

Matching( f(1, 2, 3), f(AnySequence) )True

Matching( f(x, y), f(AnySequence(Symbol)) )True

AnySequence can used in an assignment as a pattern for a sequence of arguments.

f(??Real) = "reals"; { f(1), f(1.0), f(1.0, 2.0), f(1, 0.5) }{f(1), "reals", "reals", f(1, 0.5)}

Apply@@function

Syntax

Apply(h, exp)h @@ exp

replaces the head of expression exp with h.

Details

After the new head is set, the whole expression is evaluated.

Examples

Apply(g, f(1, a, x))g(1, a, x)

Plus @@ {2, 3, 4}9

Apply(Times, a+b+c)a*b*c

f @@ (a+b)f(a, b)

Apply will process anonymous function as well:

Apply( #1^#2 & , {3, 2} )9

ArcCosfunction

Function Plot

Syntax

ArcCos(z)

gives the arcus cosine of z.

Details

ArcCos gives an exact result when possible.
If z is an exact numerical value, ArcCos remains unevaluated if no exact solution was found.
ArcCos gives complex results when z is not real or |z| > 1.
ArcCos has the attribute Listable.

Examples

ArcCos(0.5)1.0472

Calculate(ArcCos(1/2), 70)1.047197551196597746154214461093167628065723133125035273658314864102605

ArcCos(1.2)0.622363*I

ArcCos(2.0+3.0*I)1.00014-1.98339*I

Particular arguments and their result:

ArcCos(-1)Pi

ArcCos(0)1/2*Pi

ArcCos(1/2)1/3*Pi

ArcCos(1)0

ArcCotfunction

Function Plot

Syntax

ArcCot(z)

gives the arcus cotangent of z.

Details

ArcCot(z) = ArcTan(1/z).
ArcCot gives an exact result when possible.
If z is an exact numerical value, ArcCot remains unevaluated if no exact solution was found.
ArcCot has the attribute Listable.

Examples

ArcCot(2.0)0.463648

Calculate(ArcCot(2), 70)0.4636476090008061162142562314612144020285370542861202638109330887201979

ArcCot(2.0+3.0*I)0.160875-0.229073*I

Particular arguments and their result:

ArcCot(0)1/2*Pi

ArcCot(1)1/4*Pi

ArcCot(3^(1/2))1/6*Pi

ArcCot(Infinity)0

ArcCscfunction

Function Plot

Syntax

ArcCsc(z)

gives the arcus cosecant of z.

Details

ArcCsc(z) = ArcSin(1/z).
ArcCsc gives an exact result when possible.
If z is an exact numerical value, ArcCsc remains unevaluated if no exact solution was found.
ArcCsc gives complex results when z is not real or |z| < 1.
ArcCsc has the attribute Listable.

Examples

ArcCsc(3.0)0.339837

Calculate(ArcCsc(3), 70)0.3398369094541219370963925133917640663882446903324580714319239624899159

ArcCsc(0.2)1.5708-2.29243*I

ArcCsc(2.0+3.0*I)0.150386-0.231335*I

Particular arguments and their result:

ArcCsc(-1)-1/2*Pi

ArcCsc(0)ComplexInfinity

ArcCsc(2)1/6*Pi

ArcCsc(Infinity)0

ArCoshfunction

Function Plot

Syntax

ArCosh(z)

gives the area hyperbolic cosine of z.

Details

ArCosh gives an exact result when possible.
If z is an exact numerical value, ArCosh remains unevaluated if no exact solution was found.
ArCosh gives complex results when z is not real or z < 1.
ArCosh has the attribute Listable.

Examples

ArCosh(1.5)0.962424

Calculate(ArCosh(3/2), 70)0.9624236501192068949955178268487368462703686687713210393220363376803277

ArCosh(0.2)1.36944*I

ArCosh(2.0+3.0*I)1.98339+1.00014*I

Particular arguments and their result:

ArCosh(1)0

ArCosh(-1)I*Pi

ArCosh(0)1/2*I*Pi

ArCosh(1/2)1/3*I*Pi

ArCothfunction

Function Plot

Syntax

ArCoth(z)

gives the area hyperbolic cotangent of z.

Details

ArCoth(z) = ArTanh(1/z).
ArCoth gives an exact result when possible.
If z is an exact numerical value, ArCoth remains unevaluated if no exact solution was found.
ArCoth gives complex results when z is not real or |z| < 1.
ArCoth has the attribute Listable.

Examples

ArCoth(1.5)0.804719

Calculate(ArCoth(3/2), 70)0.8047189562170501873003796666130938197628006771342588609563239457370895

ArCoth(0.5)0.549306-1.5708*I

ArCoth(2.0+3.0*I)0.146947-0.231824*I

Particular arguments and their result:

ArCoth(0)Indeterminate

ArCoth(1)Infinity

ArCoth(Infinity)0

ArCschfunction

Function Plot

Syntax

ArCsch(z)

gives the area hyperbolic cosecant of z.

Details

ArCsch(z) = ArSinh(1/z).
ArCsch gives an exact result when possible.
If z is an exact numerical value, ArCsch remains unevaluated if no exact solution was found.
ArCsch has the attribute Listable.

Examples

ArCsch(0.5)1.44364

Calculate(ArCsch(1/2), 70)1.443635475178810342493276740273105269405553003156981558983054506520492

ArCsch(2.0+1.0*I)0.396568-0.186318*I

Particular arguments and their result:

ArCsch(0)ComplexInfinity

ArCsch(Infinity)0

ArcSecfunction

Function Plot

Syntax

ArcSec(z)

gives the arcus secant of z.

Details

ArcSec(z) = ArcCos(1/z).
ArcSec gives an exact result when possible.
If z is an exact numerical value, ArcSec remains unevaluated if no exact solution was found.
ArcSec gives complex results when z is not real or |z| < 1.
ArcSec has the attribute Listable.

Examples

ArcSec(3.0)1.23096

Calculate(ArcSec(3), 70)1.230959417340774682134929178247987375710340009355094839055548333663992

ArcSec(0.2)2.29243*I

ArcSec(2.0+3.0*I)1.42041+0.231335*I

Particular arguments and their result:

ArcSec(-1)Pi

ArcSec(0)ComplexInfinity

ArcSec(2)1/3*Pi

ArcSec(Infinity)1/2*Pi

ArcSinfunction

Function Plot

Syntax

ArcSin(z)

gives the arcus sine of z.

Details

ArcSin gives an exact result when possible.
If z is an exact numerical value, ArcSin remains unevaluated if no exact solution was found.
ArcSin gives complex results when z is not real or |z| > 1.
ArcSin has the attribute Listable.

Examples

ArcSin(0.5)0.523599

Calculate(ArcSin(1/2), 70)0.5235987755982988730771072305465838140328615665625176368291574320513027

ArcSin(1.2)1.5708-0.622363*I

ArcSin(2.0+3.0*I)0.570653+1.98339*I

Particular arguments and their result:

ArcSin(-1)-1/2*Pi

ArcSin(0)0

ArcSin(1/2)1/6*Pi

ArcSin(1)1/2*Pi

ArcTanfunction

Function Plot

Syntax

ArcTan(z)

gives the arcus tangent of z.

ArcTan(x,y)

gives the arcus tangent of y/x, taking into account the quadrant of the point (x, y).

Details

ArcTan gives an exact result when possible.
If z is an exact numerical value, ArcTan remains unevaluated if no exact solution was found.
ArcTan has the attribute Listable.

Examples

ArcTan(2.0)1.10715

Calculate(ArcTan(2), 70)1.107148717794090503017065460178537040070047645401432646676539207433710

ArcTan(2.0+3.0*I)1.40992+0.229073*I

Particular arguments and their result:

ArcTan(0)0

ArcTan(1)1/4*Pi

ArcTan(3^(1/2))1/3*Pi

ArcTan(Infinity)1/2*Pi

ArcTan with two arguments:

ArcTan(1, 1)1/4*Pi

ArcTan(-1, 0)Pi

ArcTan(-1.0, -3.0)-1.89255

ArcTan(-2, 1)Pi - ArcTan(1/2)

Argfunction

Function Plot

Syntax

Arg(z)

gives the argument of the (complex) number z.

Details

Arg is suitable for both symbolic and numerical manipulation.
Arg gives the phase angle in radians between -Pi and Pi.
Arg is 0 for positive non-complex numbers, and Pi for negative non-complex numbers.
Arg(0) is 0.
For special cases of z, some simplifications are performed.

Examples

Arg(1.2 + 3.4*I)1.2315

Arg(-123)Pi

Arg(1+I)1/4*Pi

Arg(2.5)0

Simplifications are performed if possible.

Arg(-2*x)Arg(-x)

Arg(x^(1/3))1/3*Arg(x)

Arg(Abs(z))0

ArSechfunction

Function Plot

Syntax

ArSech(z)

gives the area hyperbolic secant of z.

Details

ArSech(z) = ArCosh(1/z).
ArSech gives an exact result when possible.
If z is an exact numerical value, ArSech remains unevaluated if no exact solution was found.
ArSech gives complex results when z is not real or z < 0 or z > 1.
ArSech has the attribute Listable.

Examples

ArSech(0.5)1.31696

Calculate(ArSech(1/2), 70)1.316957896924816708625046347307968444026981971467516479768472256920460

ArSech(1.5)0.841069*I

ArSech(2.0+1.0*I)0.215612-1.16921*I

Particular arguments and their result:

ArSech(0)Infinity

ArSech(1)0

ArSech(2)1/3*I*Pi

ArSech(Infinity)1/2*I*Pi

ArSinhfunction

Function Plot

Syntax

ArSinh(z)

gives the area hyperbolic sine of z.

Details

ArSinh gives an exact result when possible.
If z is an exact numerical value, ArSinh remains unevaluated if no exact solution was found.
ArSinh gives complex results when z is not real.
ArSinh has the attribute Listable.

Examples

ArSinh(1.5)1.19476

Calculate(ArSinh(3/2), 70)1.194763217287109304111930828519090523536162075153005429270680299461324

ArSinh(2.0+3.0*I)1.96864+0.964659*I

Particular arguments and their result:

ArSinh(0)0

ArSinh(1*I)1/2*I*Pi

ArSinh(1/2*I)1/6*I*Pi

ArTanhfunction

Function Plot

Syntax

ArTanh(z)

gives the area hyperbolic tangent of z.

Details

ArTanh gives an exact result when possible.
If z is an exact numerical value, ArTanh remains unevaluated if no exact solution was found.
ArTanh gives complex results when z is not real or |z| > 1.
ArTanh has the attribute Listable.

Examples

ArTanh(0.5)0.549306

Calculate(ArTanh(1/2), 70)0.5493061443340548456976226184612628523237452789113747258673471668187471

ArTanh(1.5)0.804719-1.5708*I

ArTanh(2.0+3.0*I)0.146947+1.33897*I

Particular arguments and their result:

ArTanh(0)0

ArTanh(1)Infinity

ArTanh(I)1/4*I*Pi

ArTanh(-Infinity)1/2*I*Pi

Attributesfunction

Syntax

Attributes(s)

gives a list of all set attributes for symbol s.

Attributes(s) = attr

sets the attribute attr for symbol s.

Attributes(s) = {attr₁, attr₂, …}

sets the attributes attr_i for symbol s.

Details

During the set of attributes all old attributes are removed.

The following set of attributes can be used for a symbol f:

Flat	`f` is associative.
Orderless	`f` is commutative.
Listable	`f` is automatically threaded over List{…}.
Protected	Definitions of `f` cannot be changed.
Locked	Attributes of `f` cannot be changed.
HoldAll	All argument of `f` maintain unevaluated.
HoldFirst	The first argument of `f` maintains unevaluated.
HoldRest	All argument of `f`, except the first one, maintain unevaluated.
SequenceHold	An expicit Sequence as an argument of `f` maintains unevaluated.

Examples

Attributes(Plus){Flat, Listable, Locked, NumericFunction, OneIdentity, Orderless, Protected}

Define an orderless and flat expression:

f(1, z, f(a, 2))f(1, z, f(a, 2))

Attributes(f) = {Orderless, Flat}{Flat, Orderless}

f(1, z, f(a, 2))f(1, 2, a, z)

Define a function that threads over lists:

Attributes(g) = Listable; g(x?) := "<" ~ ToString(x) ~ ">"Null

g(7)"<7>"

g({-1,2,a}){"<-1>", "<2>", "<a>"}

Bernoullifunction

Syntax

Bernoulli(n)

gives the Bernoulli number B_n.

Details

The evaluation of B_n uses a summation series by Braun–Romberger–Bentz.
Bernoulli has the attribute Listable.

Examples

Bernoulli(2)1/6

Bernoulli(10)5/66

Bernoulli(60)-1215233140483755572040304994079820246041491/56786730

Possible Issues

Bernoulli is left unevaluated for non-integer or negative numbers.

Bernoulli(-2)Bernoulli(-2)

Bernoulli(3.5)Bernoulli(3.5)

Reference

Braun J, Romberger D, Bentz H J (2017), "Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers", Notes on Number Theory and Discrete Mathematics 23(2): 54–80

BinaryFormatoption

Syntax

BinaryFormat -> type

is an option to define the binary format for numeric and string representations.

Details

BinaryFormat can be used in ToString, ToInteger, and ToReal.

type is a string and can be one of the following formats:

Integer formats:
`"Integer8"`, `"int8"`	8-bit signed integer
`"Integer16"`, `"int16"`	16-bit signed integer ¹⁾
`"Integer32"`, `"int32"`	32-bit signed integer ¹⁾
`"Integer64"`, `"int64"`	64-bit signed integer ¹⁾
`"IntegerN"`, `"intn"`	n-bit signed integer ^{1) 2)}
`"UnsignedInteger8"`, `"uint8"`	8-bit unsigned integer
`"UnsignedInteger16"`, `"uint16"`	16-bit unsigned integer ¹⁾
`"UnsignedInteger32"`, `"uint32"`	32-bit unsigned integer ¹⁾
`"UnsignedInteger64"`, `"uint64"`	64-bit unsigned integer ¹⁾
`"UnsignedIntegerN"`, `"uintn"`	n-bit unsigned integer ^{1) 2)}
Floating-point formats:
`"Real16"`, `"float16"`, `"half"`	16-bit IEEE 754 floating point number
`"Real32"`, `"float32"`, `"single"`	32-bit IEEE 754 floating point number
`"Real64"`, `"float64"`, `"double"`	64-bit IEEE 754 floating point number
`"Complex32"`	Two 16-bit IEEE 754 floating point numbers
`"Complex64"`	Two 32-bit IEEE 754 floating point numbers
`"Complex128"`	Two 64-bit IEEE 754 floating point numbers
String formats:
`"ANSI"`	Windows codepage 1252 - latin 1 encoded string
`"UTF8"`	UTF-8 encoded string
`"UTF16"`	UTF-16 encoded string ¹⁾
`"UTF32"`	UTF-32 encoded string ¹⁾
¹⁾ By default the order of bytes are handled as little-endian. Use Endianness to change it.
²⁾ The size depends on the most significant bit and is rounded to a full byte.

Examples

Handling different number formats:

ToString(12345, BinaryFormat->"Integer16") ToString(3.1415, BinaryFormat->"double")'90' 'o\x12\x83ÀÊ!\t@'

ToInteger('ÿ', BinaryFormat->"Integer8") ToInteger('ÿ', BinaryFormat->"UnsignedInteger8")-1 255

Handling different string formats:

ToString("yä€𝄞", BinaryFormat->"ANSI") ToString("yä€𝄞", BinaryFormat->"UTF8") ToString("yä€𝄞", BinaryFormat->"UTF16", Endianness->-1) ToString("yä€𝄞", BinaryFormat->"UTF16", Endianness->1) ToString("yä€𝄞", BinaryFormat->"UTF32")'yä\x80?' 'yÃ¤â\x82¬ð\x9D\x84\x9E' 'y\0ä\0¬ 4Ø\x1EÝ' '\0y\0ä ¬Ø4Ý\x1E' 'y\0\0\0ä\0\0\0¬ \0\0\x1EÑ\x01\0'

ToString('yÃ¤â\x82¬ð\x9D\x84\x9E', BinaryFormat->"UTF8") ToString('\0y\0ä ¬Ø4Ý\x1E', BinaryFormat->"UTF16", Endianness->1)"yä€𝄞" "yä€𝄞"

Possible Issues

An error is given, if the format is insufficient for the whole number or its precision.

ToString(12345, BinaryFormat->"Integer8") ToString(12345, BinaryFormat->"float16")ToString.InsufficientBinaryFormat: Specified format Integer8 is insufficient to represent the whole number 12345 or its precision.ToString.InsufficientBinaryFormat: Specified format float16 is insufficient to represent the whole number 12345 or its precision.'9' '\ar'

ToReal('\ar', BinaryFormat->"float16")12344.0

Binomialfunction

Syntax

Binomial(n, k)

gives the binomial coefficient n choose k.

Details

Binomial(n, k) is defined via Factorial! as n!k! (n−k)!.
Binomial is processing non-integer values too.
Binomial has the attribute Listable.

Examples

Binomial(49, 6)13983816

Binomial(365, 31)884481797664650696626118102000059032909354320

Binomial(7/2, 3/2)35/8

Binomial(3.5, 2.1)4.26099

Binomial keeps partial products as small as possible to avoid overflows.

(10^20)!Integer.Overflow: Integer overflow during computation with integers detected.Overflow

Binomial(10^20, 3)166666666666666666661666666666666666666700000000000000000000

Boolefunction

Syntax

Boole(bool)

gives 0 if bool is True and 1 if bool is False.

Details

Boole is effectively equivalent to If(bool, 1, 0).
Boole remains unevaluated if bool is neither True nor False.
Boole has the attribute Listable.

Examples

Boole(False)0

Boole(32 > 30)1

Boole(a<b)Boole(a < b)

Boole({True, False, False}){1, 0, 0}

Calculatefunction

Syntax

Calculate(exp)

evaluates the expression exp to a real or complex number with machine precision.

Calculate(exp, n)

evaluates the expression exp to a real or complex number with precision n.

Details

Usually, expressions with exclusively exact values, will keep exact results.
Use Calculate to receive an approximate numerical value of them.
Calculate, with a specified precision n, uses internally a higher precision to ensure the required precision of the result.
Calculate do not affect the exponent of Power^ to keep fast evaluation of rational powers still possible.

Examples

Log(-5)I*Pi + Log(5)

Log(-5) // Calculate1.60944+3.14159*I

Calculate a set of expressions:

x = { 4/6 , Cos(3) , Cos(Pi/6) }{2/3, Cos(3), 1/2*3^(1/2)}

Calculate(x){0.666667, -0.989992, 0.866025}

Calculate any exact expressions with a specified precision:

Calculate(Log(-5), 25)1.609437912434100374600759+3.141592653589793238462643*I

Calculate(Cos(2^10), 50)0.98735361821984829524651338092244419111051208085007

Possible Issues

It is not possible to increase the precision of an inexact number:

x = Calculate(Pi, 15)3.14159265358979

Calculate(x, 30)3.14159265358979

In some cases the internally higher precision is not enough to ensure the required precision of the result:

y = Calculate(Sin(10^40), 30)Calculate.ExtraPrecisionLimit: Internal extra precision limit (32) reached during calculation.-0.569633400953636327308

y // Precision21.6803

A subtraction of almost equal numbers can can lead to strong loss of precision.

Calculate(Exp(1/10^100)-1, 30)Calculate.ExtraPrecisionLimit: Internal extra precision limit (32) reached during calculation.0.e-61

The exponent of Power^ is not affected by Calculate, the base is affected.

Calculate(z^4)z^4

Calculate(Pi^z)3.14159^z

Ceilfunction

Function Plot

Syntax

Ceil(x)

gives the smallest integer greater than or equal to x.

Details

If x is an exact numerical expression, the numerical approximation of x is used to establish the result.
For a complex number x, Ceil is applied separately to the real and imaginary part.
Ceil has the attribute Listable.

Examples

Ceil(2.1)3

Ceil(-2.9)-2

Ceil(3/8)1

Ceil(1.2 + 3.8*I)2+4*I

Ceil(Log(10))3

Ceil(Pi*10^50)314159265358979323846264338327950288419716939937511

Processing of Overflow and Underflow:

Ceil(Overflow)Overflow

Ceil(Underflow)1

Ceil(-Underflow)0

Clearfunction

Syntax

Clear(s)

removes all assignment rule defined to symbol s.

Clear({s₁, s₂, … })

removes all assignment rule defined to the symbols s_i.

Details

Clear can be used to remove assignments defined by Define=, DefineDelayed:=, SubDefine@=, and SubDefineDelayed@:=.
Clear does not clear the symbol's attributes. ClearAttributes have to be used, too.

Examples

Remove an assignment from a single symbols:

x = 123123

x^215129

Clear(x)Null

x^2x^2

Remove all assignments:

f(10) = 0 f(z?Integer) := 2*z f(z?) * w? @:= -w*z0

Definitions(f)f(10) = 0f(Pattern(z, ?Integer)) := 2*zf(Pattern(z, Any))*Pattern(w, Any) @:= Minus(w*z)Null

Clear(f)Null

Definitions(f)/* no definitions */Null

ClearAttributesfunction

Syntax

ClearAttributes(s)

removes all attributes for symbol s.

ClearAttributes(s, attr)

removes the attribute attr for symbol s.

ClearAttributes(s, {attr₁, attr₂, … } )

removes the attributes attr_i for symbol s.

Details

ClearAttributes gives a list of removed attributes.
See Attributes for available attributes attr.

Examples

Remove the attribute Listable of Boole:

Boole({True, False, True}){1, 0, 1}

ClearAttributes(Boole, Listable){Listable}

Boole({True, False, True})Boole({True, False, True})

However, Boole has still other attributes:

Attributes(Boole){Protected}

Possible Issues

Attributes of locked symbols can't be removed.

ClearAttributes(Plus)Attributes.Locked: Plus is locked!{}

Complementfunction

Syntax

Complement(list, list₁, list₂, … )

gives a sorted list of all distinct elements that appear in list but not in any of list_i.

Details

If the lists list and list_i are considered as sets, Complement gives their complement list \ list₁.
Complement works with any kind of expressions as long as they have the same head.

Examples

Complement( {4,2,3,2,1} , {1,4} ){2, 3}

Complement( {x, y, z} , {y} ){x, z}

Complement( {1,2,3,4} , {1,2} , {2,3} ){4}

Complement works with any expressions, not only lists, as long as they have the same head.

Complement( a + b + c + d , a + c )b + d

Complement( f(1, 4, 2), f(2, 3) )f(1, 4)

Possible Issues

Heads have to be the same when Complement is applied on expressions.

Complement( f(1, 4, 2), g(2, 3) )Complement.Heads: Heads f and g are not the same.Complement(f(1, 4, 2), g(2, 3))

Complex`x`+`y`*Iobject

Syntax

Complex(x, y)x + y*I

is a complex number with a real part x and imaginary part y multiplied with the imaginary unit I.

Complex

is the head of a complex number.

Details

Both parts x and y can be of the type Integern, Rationala/b or Realx.
Complex numbers are exact numbers as long as both parts are exact numbers.
If either the real part or the imaginary part is an inexact number, the other part is not effected by this inaccuracy.
ComplexOverflow and ComplexUnderflow are two special complex numbers representing a float pointing overflow or underflow, respectively.

Examples

1+2*I1+2*I

Complex(1,2)1+2*I

Head(1+2*I)Complex

(1-I)^3-2-2*I

1/2 + 0.4*I1/2+0.4*I

Many functions can handle and return complex numbers:

Sqrt(-4)2*I

Log(-0.5)-0.693147+3.14159*I

Exp(1+1.5*I)0.192284+2.71147*I

Abs(3+4*I)5

Possible Issues

Combinations of ComplexOverflow and ComplexUnderflow can result in Indeterminate.

ComplexOverflow * ComplexUnderflowIndeterminate

ComplexInfinityconstant

Syntax

ComplexInfinity

represents the mathematical infinite quantity with an undetermined complex phase.

Details

ComplexInfinity is the short form of DirectedInfinity().
ComplexInfinity is handled in calculations as an exact number.
ComplexInfinity keeps exact computation possible, unlike ComplexOverflow.
The magnitude of ComplexInfinity is even larger than ComplexOverflow.

Examples

Abs(ComplexInfinity) > Abs(ComplexOverflow)True

Arithmetical and mathematical functions are able to process ComplexInfinity:

1 / ComplexInfinity0

1 / 0ComplexInfinity

Log(ComplexInfinity)Infinity

ComplexOverflowconstant

Syntax

ComplexOverflow

represents an overflow in float-pointing arithmetic with complex numbers.

Details

The head of ComplexOverflow is Complex.
The magnitude of ComplexOverflow is always larger than the highest possible magnitude of a complex value, but always smaller than ComplexInfinity.

Examples

(7.0+4.0*I) ^ Exp(100.0)ComplexOverflow

Arithmetical and mathematical functions are able to process ComplexOverflow:

3.14 + ComplexOverflowComplexOverflow

ComplexOverflow * (-2)ComplexOverflow

Abs( ComplexOverflow )Overflow

Arg( ComplexOverflow )Indeterminate

ComplexUnderflowconstant

Syntax

ComplexUnderflow

represents an underflow in float-pointing arithmetic with complex numbers.

Details

The head of ComplexUnderflow is Complex.
The magnitude of ComplexUnderflow is always smaller than the lowest possible magnitude of a complex value, but always larger than 0.

Examples

(7.0+4.0*I) ^ (-Exp(100.0))ComplexUnderflow

Arithmetical and mathematical functions are able to process ComplexUnderflow:

3.14 + ComplexUnderflow3.14+0.0*I

ComplexUnderflow * (-2)ComplexUnderflow

1 / ComplexUnderflowComplexOverflow

Abs( ComplexUnderflow )Underflow

Arg( ComplexUnderflow )Indeterminate

Compound;function

Syntax

Compound(exp₁, exp₂, … )exp₁ ; exp₂ ; …

evaluates all expressions exp_i in turn and gives the last one as result.

exp;

evaluates exp but gives Null as result.

Details

Compound is usually used in expressions to perform multiple calculations.

Examples

x = 1 ; y = 2 ; x + y3

z = 10^10000;Null

Log(10, z)10000

Condition/?function

Syntax

Condition(pat, cond)pat /? cond

is a pattern object that matches only if the evaluation of cond yields True.

Details

The condition is only evaluated if the pattern pat matches at all.
Condition can be applied to a single pattern object or to a whole expression.
In a collection of definitions, the next definition is tried if cond yields not True.
That distinguishes the use of Condition/? instead of If.

Examples

Matching( 1, x? /? x>0 ) Matching( -1, x? /? x>0 )True False

Using Condition in a single argument:

f(x? /? x!=0) := 1/xNull

f(2) f(1/2) f(0)1/2 2 f(0)

Using Condition for the whole definition:

g(x?, y?) /? x>y := x^yNull

g(10, 2) g(2, 4) g(x, 3)100 g(2, 4) g(x, 3)

Using Condition for a replacement rule:

Replace( {-1,2,-3,4,-5,6} , x?Integer /? x<0 -> 0 ){0, 2, 0, 4, 0, 6}

The difference between a definition with Condition/? instead of a definition with If:

g( x? /? x!=1 ) := (x^2-1)/(x-1) /* general definition with exception */ h(x?) := If( x!=1 , (x^2-1)/(x-1) , "problem" ) /* definition with If */Null

{ g(0.0) , g(1.0) , g(2.0) } { h(0.0) , h(1.0) , h(2.0) }{1.0, g(1.0), 3.0} {1.0, "problem", 3.0}

Conjugatefunction

Syntax

Conjugate(z)

gives the complex conjugate of the complex number z.

Details

Conjugate flips the sign of the imaginary part.
Conjugate has the attribute Listable.
For special cases of z, some simplifications are performed.

Examples

Conjugate(1+I)1-I

Conjugate(1.2 - 3.4*I)1.2+3.4*I

Conjugate(-123)-123

Conjugate(I*Infinity)DirectedInfinity(-I)

Simplifications are performed if possible.

Conjugate(-2*x)-2*Conjugate(x)

Conjugate(Log(3+I))Log(3-I)

Conjugate((-2)^(1/2))-I*2^(1/2)

Containfunction

Syntax

Contain(exp, el)

gives True if exp contains el at any level, and False otherwise.

Details

Usually, exp is a List{…} and Contain verifies if el is a member.
However, exp can any kind of expression and el can be a pattern object as well.

Examples

Working with lists:

Contain( {1,2,3,4,5} , 3 ) Contain( {1,2,3,4,5} , 6 )True False

Contain( {1, 2, {31, 32}, 4} , 31 )True

Contain( {1, Pi, 4.5, f()} , ?Real)True

Working with expressions:

Contain( 3+4*x^2, x)True

Contain( 2 + 2*x + 3*x^2 , Condition(x^y?, y>3) ) Contain( 2 + 2*x + 3*x^2 - 5*x^4 , Condition(x^y?, y>3) )False True

Cosfunction

Function Plot

Syntax

Cos(z)

gives the cosine of z.

Details

Cos gives an exact result when possible.
If z is an exact numerical value, Cos remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Cos has the attribute Listable.

Examples

Cos(2.0)-0.416147

Calculate(Cos(2), 70)-0.4161468365471423869975682295007621897660007710755448907551499737819649

Cos(2.0+3.0*I)-4.18963-9.10923*I

Particular arguments and their result:

Cos(0)1

Cos(Pi/2)0

Cos(Pi/3)1/2

Cos(Pi/4)1/2^(1/2)

Pure imaginary arguments are converted into Cosh:

Cos(2*I)Cosh(2)

The cosine of the arcus cosine is identity:

Cos(ArcCos(z))z

Coshfunction

Function Plot

Syntax

Cosh(z)

gives the hyperbolic cosine of z.

Details

Cosh gives an exact result when possible.
If z is an exact numerical value, Cosh remains unevaluated if no exact solution was found.
Cosh has the attribute Listable.

Examples

Cosh(2.0)3.7622

Calculate(Cosh(2), 70)3.762195691083631459562213477773746108293973558230711602777643347588324

Cosh(2.0+3.0*I)-3.72455+0.511823*I

Particular arguments and their result:

Cosh(0)1

Pure imaginary arguments are converted into Cos:

Cosh(2*I)Cos(2)

The hyperbolic cosine of the area hyperbolic cosine is identity:

Cosh(ArCosh(z))z

Cotfunction

Function Plot

Syntax

Cot(z)

gives the cotangent of z.

Details

Cot(z) = 1 / Tan(z).
Cot gives an exact result when possible.
If z is an exact numerical value, Cot remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Cot has the attribute Listable.

Examples

Cot(2.0)-0.457658

Calculate(Cot(2), 70)-0.4576575543602857637502774104320472764284863292316743296413921626362923

Cot(2.0+3.0*I)-0.00373971-0.996758*I

Particular arguments and their result:

Cot(0)ComplexInfinity

Cot(Pi/2)0

Cot(Pi/3)1/3^(1/2)

Cot(Pi/4)1

The tangent of the arcus tangent is identity:

Cot(ArcCot(z))z

Cothfunction

Function Plot

Syntax

Coth(z)

gives the hyperbolic cotangent of z.

Details

Coth(z) = 1 / Tanh(z).
Coth gives an exact result when possible.
If z is an exact numerical value, Coth remains unevaluated if no exact solution was found.
Coth has the attribute Listable.

Examples

Coth(1.0)1.31304

Calculate(Coth(1), 70)1.313035285499331303636161246930847832912013941240452655543152967567084

Coth(1.0+2.0*I)0.82133+0.171384*I

Particular arguments and their result:

Coth(0)ComplexInfinity

Coth(Infinity)1

The hyperbolic cotangent of the area hyperbolic cotangent is identity:

Coth(ArCoth(z))z

Countfunction

Syntax

Count(exp, el)

gives the number of elements in exp that match el.

Count(str, sr)

gives the number of sr occurring in str.

Details

Usually, exp is a List{…} and Count gives the nummer of members matching el.
However, exp can any kind of expression and el can be a pattern object as well.
If only the presence have to be checked, instead of exact number of ocurres, Contain is faster than Count.

Examples

Working with lists:

Count( {1, 0, 0, 1, 0} , 1 )2

Count( {a, x, {b, x}, x, c} , x )3

Count( {1, Pi, 4.5, f(), 2.0} , ?Real)2

Working with expressions:

Count( 3+3*x+4*x^2, x)2

Count( 2 + 2*x + 3*x^2 , x^y?)1

Working with strings:

Count( "foobaabaabaafoofoobaa", "baa")4

Count( 'ABAABABAA', 'A')6

Possible Issues

Count uses Matching for comparison and no mathematically equallity.

Count( {2.0, 2.0, 3.0} , 2 ) Count( {2.0, 2.0, 3.0} , 2.0 )0 2

Cscfunction

Function Plot

Syntax

Csc(z)

gives the cosecant of z.

Details

Csc(z) = 1 / Sin(z).
Csc gives an exact result when possible.
If z is an exact numerical value, Csc remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Csc has the attribute Listable.

Examples

Csc(2.0)1.09975

Calculate(Csc(2), 70)1.099750170294616466756697397026312896658764443149845708742554443062569

Csc(2.0+3.0*I)0.0904732+0.041201*I

Particular arguments and their result:

Csc(0)ComplexInfinity

Csc(Pi/2)1

Csc(Pi/3)2/3^(1/2)

Csc(Pi/4)1/2^(1/2)

The tangent of the arcus tangent is identity:

Csc(ArcCsc(z))z

Cschfunction

Function Plot

Syntax

Csch(z)

gives the hyperbolic cosecant of z.

Details

Csch(z) = 1 / Sinh(z).
Csch gives an exact result when possible.
If z is an exact numerical value, Csch remains unevaluated if no exact solution was found.
Csch has the attribute Listable.

Examples

Csch(2.0)0.275721

Calculate(Csch(2), 70)0.2757205647717832077583514821630271212496226719912580519731712337225655

Csch(2.0+3.0*I)-0.272549-0.0403006*I

Particular arguments and their result:

Csch(0)ComplexInfinity

Csch(Infinity)0

The hyperbolic secant of the area hyperbolic secant is identity:

Csch(ArCsch(z))z

Define=function

Syntax

Define(lhs, rhs)lhs = rhs

evaluates the expression rhs and assign the result as a replacement for the expression lhs whenever it appears.

Details

lhs is typically a symbol or an expression with pattern objects.
For Define the assignment f(g) = rhs is associate with f, while f(g) @= rhs (SubDefine@=) is associate with g.
The expression rhs will evaluated before assignment.
Multiple assignments to the same expression head are ordered by their specialization.
If lhs and rhs are lists with same length, the assignment is done with corresponding elements.
lhs can contain Condition/? to specialize the definition.
Clear can be used to remove all assignment rules from a symbol.

Examples

Assign expressions to single symbols:

x = 123123

x*2246

y = a^2 + b^2a^2 + b^2

2*y2*(a^2 + b^2)

Multiple assignments via list:

{a, b, c} = {123, 3.14, "Test"};Null

a b c123 3.14 "Test"

Assign expressions to an expression with a pattern:

f(z?) = z^2 + zz + z^2

f(9) f(a+b)90 16037.4

Polytypism allows multiple definitions to the same expression head:

f(z?, n?) = z^n + zz + z^n

f(w) f(w, 5)w + w^2 w + w^5

Possible Issues

Define evaluates the expression rhs on the right hand side before assignment, DefineDelayed:= not.

x = 3.143.14

h(x?) = 2*x6.28

k(x?) := 2*xNull

x is already known and will be replaced in Define= but not in DefineDelayed:=.

h(123) k(123)6.28 246

DefineDelayed:=function

Syntax

DefineDelayed(lhs, rhs)lhs := rhs

assigns the unevaluated expression rhs as a replacement for the expression lhs whenever it appears.

Details

lhs is typically a symbol or an expression with pattern objects.
For DefineDelayed the assignment f(g) := rhs is associate with f, while f(g) @:= rhs (SubDefineDelayed@:=) is associate with g.
The expression rhs will assigned unevaluated and is always evaluated when lhs is called.
Multiple assignments to the same expression head are ordered by their specialization.
If lhs and rhs are lists with same length, the assignment is done with corresponding elements.
lhs can contain Condition/? to specialize the definition.
Clear can be used to remove all assignment rules from a symbol.

Examples

Assign expressions to single symbols:

x := 123Null

x*2246

y := a^2 + b^2Null

2*y2*(a^2 + b^2)

Multiple assignments via list:

{a, b, c} := {123, 3.14, "Test"}Null

a b c123 3.14 "Test"

Assign expressions to an expression with a pattern:

f(z?) := z^2 + zNull

f(9) f(a+b)90 16037.4

Polytypism allows multiple definitions to the same expression head:

f(z?, n?) := z^n + zNull

f(w) f(w, 5)w + w^2 w + w^5

Possible Issues

DefineDelayed does not evaluates the expression rhs on the right hand side before assignment, Define= does.

m = 22

k(q?) := m*qNull

h(q?) = m*q2*q

m is already known and will be replaced within Define= during the assignment but for DefineDelayed:= after replacement.

h(a) k(a)246 246

The result of h(a) changes when m is changed, but not the result of k(a)

m = 55

h(123) k(123)246 615

Definitionsfunction

Syntax

Definitions(s)

prints all assignments defined to symbol s.

Details

For built-in functions just the assignment pattern is printed.
The definitions are printed in order of their specialization, started with the most specialized one.

Examples

Definitions(Plus)Plus(AnyNullSequence) /* built-in function */Null

Define some assignments to f and print all definitions:

f(1) = 1; f(x?) := f(x-1)+f(x-2); f(2) = 1; f(x? /? x <= 0) = 0;Null

Definitions(f)f(1) = 1f(2) = 1f(Condition(Pattern(x, Any), x <= 0)) = 0f(Pattern(x, Any)) := f(x - 1) + f(x - 2)Null

Denominatorfunction

Syntax

Denominator(z)

gives the denominator of the number z.

Details

For a rational number z = a/b, Denominator gives b.
For integer and real numbers Numerator(z) = 1.
Denominator picks out terms with a negative exponent.
Denominator has the attribute Listable.

Examples

Denominator(3/7)7

Denominator(-8/3)3

Denominator(123)1

Denominator(3.4)1

Denominator(1/2 + 1/3*I)6

Denominator(f/g)g

Denominator(a^2*b^(-3)*c^4)b^3

Depthfunction

Syntax

Depth(exp)

gives the maximum nesting depth of exp.

Details

For a non-expression objects, Depth is 0.

Examples

Depth( { {1,2}, 3, {4,{5,6}} } )3

Depth( 123 )0

Depth works with any expressions, not only lists.

Depth( f(x^2) )2

Depth( a + 2*b^2 + 3*c )3

Rational and complex numbers have a depth of 0.

Depth( 1/3 )0

Depth( 1+2*I )0

Possible Issues

Depth operates on the internal evaluated form, not on the displayed form.

a/b Depth( a/b )a/b 2

InternalForm( a/b )Times(a, Power(b, -1))

DigitBaseoption

Syntax

DigitBase -> int

is an option to define the digit base for number representations.

Details

DigitBase can be used in ToString, ToInteger, and ToReal.
int has to be an integer between 2 and 36.
If DigitBase is not explicitly specified, the base 10 is used.
For bases larger than ten, the capital letters from A until Z are used.
During parsing also the small letters from a until z are valid.

Examples

ToString(12345, DigitBase->2)"11000000111001"

ToString(10^100, DigitBase->36)"2HQBCZU2OW52BALA8LGC3S5Y9MM5TIY0VO9TKE25466GFI6AX8GS22X7KUU8L1TDS"

ToInteger("FF", DigitBase->16)255

ToInteger("5yc1s", DigitBase->36)10000000

ToReal("0.C4", DigitBase->16)0.765625

DirectedInfinityfunction

Syntax

DirectedInfinity()

represents the mathematical infinite quantity with an unknown direction in the complex plane.

DirectedInfinity(z)

represents the mathematical infinite quantity with its direction in the complex plane along z.

Details

During evaluation z will be normalized.
DirectedInfinity(1) is displayed as Infinity.
DirectedInfinity(-1) is displayed as -Infinity.
DirectedInfinity(0) is converted to DirectedInfinity().
DirectedInfinity() is displayed as ComplexInfinity.

Examples

Arithmetical and mathematical functions are able to process DirectedInfinity:

DirectedInfinity(1.0+I) * DirectedInfinity(I)DirectedInfinity(-0.707107+0.707107*I)

DirectedInfinity(I) ^ 4Infinity

Tan(DirectedInfinity(1+I))I

ArcCos(Infinity)DirectedInfinity(I)

ArTanh(DirectedInfinity(I))1/2*I*Pi

Divide/function

Syntax

Divide(exp₁, exp₂)exp₁ / exp₂

divides expression exp₁ by exp₂.

Details

Divide(exp₁, exp₂) is converted to Times(exp₁, Power(exp₂, -1)).
Divide gives a rational number in case of two non-divisible integer numbers.
Divide is suitable for both symbolic and numerical manipulation.
Divide has the attribute Listable.

Examples

4 / 22

x^5 / x^2x^3

Possible Issues

A division of two exact numbers keeps exact. Use at least one real number to force a real result.

4 / 6 4.0 / 62/3 0.666667

Econstant

Syntax

E

represents Euler's number e ≅ 2.71828.

Details

E is exact and is not evaluated unless it is combined with an inexact computation.
E can be calculated with arbitrary precision.
For calculations with an arbitrary precision the factorial series is used.

Examples

Calculate E:

Calculate(E)2.71828

Calculate(E, 70)2.718281828459045235360287471352662497757247093699959574966967627724077

Arithmetic with E:

2*E + 3*E5*E

2.0*E5.43656

Some functions give exact results for terms with E.

Log(E)1

Endiannessoption

Syntax

Endianness -> -1Endianness -> 1

is an option to define the endianness (order of bytes) as little-endian (-1) or big-endian (1).

Details

Endianness can be used in ToString and ToInteger.
If Endianness is not explicitly specified, little-endian (-1) is used.
Endianness usually used in combination with BinaryFormat.

Examples

ToString(1023, BinaryFormat->"Integer32") ToString(1023, BinaryFormat->"Integer32", Endianness->-1) ToString(1023, BinaryFormat->"Integer32", Endianness->1)'ÿ\x03\0\0' 'ÿ\x03\0\0' '\0\0\x03ÿ'

ToString("yä€𝄞", BinaryFormat->"UTF16", Endianness->-1) ToString("yä€𝄞", BinaryFormat->"UTF16", Endianness->1)'y\0ä\0¬ 4Ø\x1EÝ' '\0y\0ä ¬Ø4Ý\x1E'

ToInteger('ÿ\x01\0\0', BinaryFormat->"Integer32") ToInteger('ÿ\x01\0\0', BinaryFormat->"Integer32", Endianness->-1) ToInteger('ÿ\x01\0\0', BinaryFormat->"Integer32", Endianness->1)511 511 -16711680

Equal==function

Syntax

Equal(exp₁, exp₂, … )exp₁ == exp₂ == …

gives True if all expressions exp_i are mathematically equal, and False if they are unequal.

Details

Equal verify the expressions for mathematical equality, unlike Same===.
Inexact numbers are considered as equal if they differ in at most their last five binary digits.

Examples

1 == 1True

3.0 == 3True

0.2 == 1/5True

The integer 1 and the real number 1.0 are mathematically equal, but not identical.

1 == 1.0 1 === 1.0True False

The equality of Overflows is unknown, but they are identical.

Overflow == Overflow Overflow === OverflowOverflow == Overflow True

Equal stays unevaluated if equality is unknown.

a == ba == b

Possible Issues

Inexact numbers are considered as equal if just the last one or two decimal digits are different. This allows an easier handling with floating point numbers.

1.0000000000000000002 == 1.0000000000000000001True

1.00000000000000000020 == 1.00000000000000000010False

Erffunction

Function Plot

Syntax

Erf(x)

gives the error function value of x.

Details

Erf gives an exact result when possible.
Erf has the attribute Listable.
To resolve small differences to 1 for large value of x, Erfc can be used.
For calculations with an arbitrary precision the Taylor series and continued fraction is used.

Examples

Erf(1.0)0.842701

Calculate(Erf(1), 70)0.8427007929497148693412206350826092592960669979663029084599378978347173

Erf(-0.2)-0.222703

Erf(3.0)0.999978

Particular arguments and their result:

Erf(0)0

Erf(Infinity)1

Possible Issues

Erf rapidly reaches 1.0 for positive numbers. A calculation requires higher precision to resolve the correct result.

1 - Calculate(Erf(10)) 1 - Calculate(Erf(10), 60)0.0 2.08848758376254e-45

Erfc can resolve the small difference to 1.0 without requiring higher precision.

Calculate(Erfc(10))2.08849e-45

Erfcfunction

Function Plot

Syntax

Erfc(x)

gives the complementary error function value of x.

Details

Erfc(x) = 1 - Erf(x).
Erfc gives an exact result when possible.
Erfc has the attribute Listable.
For calculations with an arbitrary precision the Taylor series and continued fraction is used.

Examples

Erfc(1.0)0.157299

Calculate(Erfc(1), 70)0.1572992070502851306587793649173907407039330020336970915400621021652827

Erfc(-2.2)1.99814

Erfc(40.0)1.896961059966e-697

Particular arguments and their result:

Erfc(0)1

Erfc(-Infinity)2

Erfc(Infinity)0

EulerMascheroniconstant

Syntax

EulerMascheroni

represents the Euler–Mascheroni constant γ ≅ 0.577216.

Details

EulerMascheroni is exact and is not evaluated unless it is combined with an inexact computation.
EulerMascheroni can be calculated with arbitrary precision.
For calculations with an arbitrary precision the Bessel function method by Brent–McMillan is used.

Examples

Calculate EulerMascheroni:

Calculate(EulerMascheroni)0.577216

Calculate(EulerMascheroni, 70)0.5772156649015328606065120900824024310421593359399235988057672348848677

Arithmetic with EulerMascheroni:

2*EulerMascheroni + 3*EulerMascheroni5*EulerMascheroni

2.0*EulerMascheroni1.15443

EulerMascheroni is the limiting difference between the harmonic series and the natural logarithm:

HarmonicNumber(10000) - Log(10000) // Calculate0.577266

References

Wikipedia: Euler's_constant
Brent R P, McMillan E M (1979), "Some New Algorithms for High-Precision Computation of Euler’s constant", Math. Comput. 34: 305–312
Gourdon X, Sebah P (2004), "The Euler constant : γ", Young 1: 2n

Expfunction

Function Plot

Syntax

Exp(z)

gives the exponential of z.

Details

Exp gives an exact result when possible.
Exp(z) is converted to Power(E, z) if no solution was found.
Exp has the attribute Listable.
For calculations with an arbitrary precision the formal power series definition is used.

Examples

Exp(-2.5)0.082085

Exp(1000.0)1.970071114017e434

Exp(3.0+2.0*I)-8.35853+18.2637*I

Calculate(Exp(10), 70)22026.46579480671651695790064528424436635351261855678107423542635522520

Particular arguments and their result:

Exp(0)1

Exp(1)E

Exp(-Infinity)0

Expandfunction

Syntax

Expand(exp)

expands out products and positive integer powers in exp.

Details

Expand uses the distributive law and the binomial theorem to expand out exp.

Examples

Expand( 3*(a+b) )3*a + 3*b

Expand( (a+b)^2 )a^2 + 2*a*b + b^2

Expand( (Pi+1)^3 * (E-2) )-2 + E - 6*Pi + 3*E*Pi - 6*Pi^2 + 3*E*Pi^2 - 2*Pi^3 + E*Pi^3

Expand( (a+b)^2 + (a-b)^2 )2*a^2 + 2*b^2

Possible Issues

Expand is not applied to deeper expression levels:

Expand( f((1+a)^2) )f((1 + a)^2)

Expression`f`(…)object

Syntax

head(arg₁, arg₂, … )

is an expression with a head head and any number of arguments arg_i.

head @ arg

is the prefix notation of an expression head(arg).

arg // head

is the postfix notation of an expression head(arg).

Details

head is usually a Symbols, but can be any other expression.
The evaluation of expressions can be controlled with attributes.
Any kind of object can be assigned to an expression via Define=, DefineDelayed:=, SubDefine@=, or SubDefineDelayed@:=.
Expressions usually contain patterns as arguments in an assignment.

Examples

Head( func(a, b) )func

Head( h(y)(a, b) )h(y)

func @ 123func(123)

123 // funcfunc(123)

Define assignments for specific arguments:

f(1) = 2 f(foo) = 3 f("bar") = 42 3 4

{ f(), f(0), f(1) } { f(foo), f(bar), f("bar"), f("foo") }{f(), f(0), 2} {3, f(bar), 4, f("foo")}

Define a function using a pattern like Any?:

g(x?) := x^2Null

{ g(7) , g(a) , g(2+b) }{49, a^2, (2 + b)^2}

Factorial!function

Function Plot

Syntax

Factorial(n)n!

gives the factorial of n.

Details

For non-integer numbers, the value of n! is given by the gamma function Γ ( 1 + n ).
The evaluation of Γ ( 1 + n ) uses Lanczos approximation and Spouge's approximation.
Factorial has the attribute Listable.

Examples

Factorial(10)3628800

50!30414093201713378043612608166064768844377641568960512000000000000

(-5)!ComplexInfinity

6.32!1321.18

1000.0!4.02387260077e2567

Calculate(Factorial(22/7), 70)7.199869288735633700776845665344668943490127819531077381591855761162949

(-1/2)!Pi^(1/2)

(2.1+0.4*I)!1.97507+0.802929*I

Falseconstant

Syntax

False

represents the boolean value false.

Details

False is a possible result of comparisons and logical operations.

Examples

1 == 2False

Not(False)True

Fibonaccifunction

Function Plot

Syntax

Fibonacci(n)

gives the Fibonacci number F_n.

Details

For integer numbers, the value of Fibonacci(n) is given by the recursion rule F_n = F_n−1 + F_n−2, with F₀ = 0 and F₁ = 1.
The evaluation of F_n uses fibonacci doubling method.
For non-integer numbers, the value of Fibonacci(n) is given by the general formula F_n = φⁿ − cos(π n) φ⁻ⁿ5, where φ is the golden ratio.
Fibonacci has the attribute Listable.

Examples

Calculate Fibonacci numbers based on the recursion rule:

Fibonacci(10)55

Fibonacci(350)6254449428820551641549772190170184190608177514674331726439961915653414425

Fibonacci(-10)-55

Calculate Fibonacci numbers based on the general formula:

Fibonacci(10.0)55.0

Fibonacci(16.32)1151.31

Fibonacci(10000.0)3.36447648764e2089

Calculate(Fibonacci(22/7), 70)2.118038120159998741207512266041453547560232382534595267287508988177487

Fibonacci(2.1+0.4*I)0.932786+0.370984*I

Flatattribute

Syntax

Flat

is an attribute for symbols to indicate that their arguments can be flatted or sub expressions can be created during pattern matching.

Details

Typical functions like Plus+ and Times* have the attribute Flat.
Attributes can be set and get with Attributes.

Examples

Summands in Plus+ are flattened during evaluation:

a + (b + c)a + b + c

By default expressions with the same head are not flattened or nested.

f(1, f(2, 3))f(1, f(2, 3))

Matching( f(1, 2, 3) , f(x?, y?) )False

After setting the attribute Flat, expressions will flattened and pattern matching tries all sub expressions.

Attributes(f) = Flat{Flat}

f(1, f(2, 3))f(1, 2, 3)

f(1, 2, 3) /: f(x?, y?) :> {x, y}{1, f(2, 3)}

Flattenfunction

Syntax

Flatten(exp)

flattens out all nested expressions having the same head as exp.

Flatten(exp, n)

flattens only down to depth level n.

Details

n has to be a non-negative integer number or Infinity.
Symbols with the attribute Flat are flatten automatically.

Examples

Flatten( f( 1, f(), 2, f(3, 4)) )f(1, 2, 3, 4)

Flatten( { a, {b1, b2}, {c1, c2, {c3x, c3y}} } ){a, b1, b2, c1, c2, c3x, c3y}

Using the depth specification:

Flatten( {1, {2, {3, {4, {5}}}}}, 1 ){1, 2, {3, {4, {5}}}}

Flatten( {1, {2, {3, {4, {5}}}}}, 2 ){1, 2, 3, {4, {5}}}

Flatten( {1, {2, {3, {4, {5}}}}}, 3 ){1, 2, 3, 4, {5}}

Flatten( {1, {2, {3, {4, {5}}}}}, 4 ){1, 2, 3, 4, 5}

Floorfunction

Function Plot

Syntax

Floor(x)

gives the greatest integer less than or equal to x.

Details

If x is an exact numerical expression, the numerical approximation of x is used to establish the result.
For a complex number x, Floor is applied separately to the real and imaginary part.
Floor has the attribute Listable.

Examples

Floor(-2.1)-3

Floor(2.9)2

Floor(7/8)0

Floor(1.2 + 3.8*I)1+3*I

Floor(Log(10))2

Floor(Pi*10^50)314159265358979323846264338327950288419716939937510

Processing of Overflow and Underflow:

Floor(Overflow)Overflow

Floor(Underflow)0

Floor(-Underflow)-1

Function&function

Syntax

Function(body)body&

is an anonymous function with the body body.

Details

body usually contains an expression with Slot# elements representing the arguments of this function.
body is not evaluated as long as the function receives no arguments.
Anonymous functions usually are used in functions like Map/@, Sort or Select.

Examples

f = Function( 2*#+1 )Function(2*#1 + 1)

f(x) f(5)1 + 2*x 11

g = #1^#2 + #2^#1 &Function(#1^#2 + #2^#1)

g(x, y) g(4,3)x^y + y^x 145

Select( {1,2,3,4} , #<3 & ){1, 2}

Possible Issues

Too many arguments are ignored. Too few arguments will result in an error.

f = {#1, #2} &Function({#1, #2})

f(a,b,c){a, b}

f(a,b){a, b}

f(a)Function.Slot: Slot number 2 cannot be filled.{a, #2}

GoldenAngleconstant

Syntax

GoldenAngle

represents the golden angle 2 πφ² ≅ 2.39996 ≅ 137.5°, with the golden ratio φ.

Details

GoldenAngle is exact and is not evaluated unless it is combined with an inexact computation.
GoldenAngle is defined as 2 πφ² = (3 − 5) π.
GoldenAngle can be calculated with arbitrary precision.

Examples

Calculate GoldenAngle:

Calculate(GoldenAngle)2.39996

Calculate(GoldenAngle, 70)2.399963229728653322231555506633613853124999011058115042935112750731307

Calculate(GoldenAngle*180/Pi)137.508

GoldenRatioconstant

Syntax

GoldenRatio

represents the golden ratio φ ≅ 1.61803.

Details

GoldenRatio is exact and is not evaluated unless it is combined with an inexact computation.
GoldenRatio is defined as φ = 1+52.
GoldenRatio can be calculated with arbitrary precision.

Examples

Calculate GoldenRatio:

Calculate(GoldenRatio)1.61803

Calculate(GoldenRatio, 70)1.618033988749894848204586834365638117720309179805762862135448622705260

Arithmetic with GoldenRatio:

GoldenRatio - 1.0/GoldenRatio1.0

Greater>function

Syntax

Greater(x₁, x₂, … )x₁ > x₂ > …

gives True if x_i is greater than x_i + 1 for all arguments, and False if x_i is less than or equal to x_i + 1 for at least one pair.

Details

For exact expressions in x_i, Greater compares the approximated value.

Examples

2 > 1True

1/3 > 0 > -0.5True

x > 2 > 2False

The comparison will be simplified if possible.

2 > 1 > x1 > x

The value of an exact expression is approximated for comparison.

Sin(2) > 0True

GreaterEqual>=function

Syntax

GreaterEqual(x₁, x₂, … )x₁ >= x₂ >= …

gives True if x_i is greater than or equal to x_i + 1 for all arguments, and False if x_i is less than x_i + 1 for at least one pair.

Details

For exact expressions in x_i, GreaterEqual compares the approximated value.

Examples

2 >= 1True

3/2 >= 1.5 >= -0.5True

x >= 1 >= 2False

The comparison will be simplified if possible.

2 >= 1 >= x1 >= x

The value of an exact expression is approximated for comparison.

Sin(2) >= 0True

HarmonicNumberfunction

Syntax

HarmonicNumber(n)

gives the n^th harmonic number H_n.

Details

H_n is given by nΣk = 1 1k .
HarmonicNumber has the attribute Listable.

Examples

HarmonicNumber(5)137/60

HarmonicNumber(60)15117092380124150817026911/3230237388259077233637600

HarmonicNumber(100) // Calculate5.18738

HarmonicNumber({1,2,3,4,5}){1, 3/2, 11/6, 25/12, 137/60}

Possible Issues

HarmonicNumber is left unevaluated for non-integer or negative numbers.

HarmonicNumber(-2)-2

HarmonicNumber(3.5)HarmonicNumber(3.5)

Headfunction

Syntax

Head(exp)

gives the head of the expression exp or any kind of object.

Details

Head can also used for non-expressional objects.
A part specification like exp[0] is a synonym for Head(exp).

Examples

Head( f(a, b) )f

Head( {1, 2, 3, 4} )List

Head( 123 )Integer

Head( x )Symbol

Head( a * b )Times

The 0^th element is the head.

g(1, 2) // Head g(1, 2)[0]g g

Possible Issues

Head operates on the internal evaluated form, not on the displayed form.

a/b Head(a/b)a/b Times

InternalForm(a/b)Times(a, Power(b, -1))

HoldAllattribute

Syntax

HoldAll

is an attribute for symbols to indicate that all arguments maintain unevaluated.

Details

HoldAll can be used to prevent the evaluation of all arguments before they are passed to the function.
Attributes can be set and get with Attributes.

Examples

Functions like And&& and Iterate have the attribute HoldAll to prevent argument evaluation.

Attributes(And){Flat, HoldAll, Protected}

Prevent the evaluation of all arguments before they are passed, to match the definition pattern:

Attributes(h) = {HoldAll}{HoldAll}

h(x?+y?) := {x, y} /* h has the attribute HoldAll */ f(x?+y?) := {x, y} /* f hasn't the attribute HoldAll */Null

h(3+7) /* 3+7 holds and matches x?+y? */ f(3+7) /* 3+7 is evaluated to 10 and does not matching */{3, 7} f(10)

HoldFirstattribute

Syntax

HoldFirst

is an attribute for symbols to indicate that the first arguments maintain unevaluated.

Details

HoldFirst can be used to prevent the evaluation of the first argument before it is passed to the function.
Attributes can be set and get with Attributes.

Examples

Functions like Define= and SubDefine@= have the attribute HoldFirst to prevent the evaluation of the symbol.

Attributes(Define){HoldFirst, Locked, Protected, SequenceHold}

Prevent the evaluation of the first argument before it is passed, to match the definition pattern:

Attributes(h) = {HoldFirst}{HoldFirst}

h(x?+y?) := {x, y} /* h has the attribute HoldFirst */ f(x?+y?) := {x, y} /* f hasn't the attribute HoldFirst */Null

h(3+7) /* 3+7 holds and matches x?+y? */ f(3+7) /* 3+7 is evaluated to 10 and does not matching */{3, 7} f(10)

HoldRestattribute

Syntax

HoldRest

is an attribute for symbols to indicate that all arguments except the first one maintain unevaluated.

Details

HoldRest can be used to prevent the evaluation of all arguments except the first one before they are passed to the function.
Attributes can be set and get with Attributes.

Examples

Functions like If have the attribute HoldRest to prevent argument evaluation, except the first one.

Attributes(If){HoldRest, Protected}

Iconstant

Syntax

I

represents the imaginary unit i ≡ −1.

Details

The imaginary part of complex numbers are represented with I.
A single I is equivalent to a complex number with a real part of 0 and an imaginary part of 1.

Examples

1 + 2*I1+2*I

I // HeadComplex

I * I-1

Sqrt(-4)2*I

Iffunction

Syntax

If(con, t, f)

gives t if con is True and f if con is False.

If(con, t, f, u)

gives u if con is neither True nor False.

If(con, t)

gives Null if con is False.

Details

Usually, con is a comparison or logical expression.
If evaluates only the argument determined by the value of the condition.

Examples

If(True, 123, -10) If(False, 123, -10)123 -10

x = 3; If(x!=0, 1.0/x, "Div 0") x = 0; If(x!=0, 1.0/x, "Div 0")0.333333 "Div 0"

If remains unevaluated if the condition is neither True nor False.

If(a==b, 1, 2)If(a == b, 1, 2)

Force a result for indefinite expressions with the fourth argument:

If(a==b, 1, 2, 3)3

Imfunction

Syntax

Im(z)

gives the imaginary part of the complex number z.

Details

If z is a real number, Im(z) is 0.
Im has the attribute Listable.
For special cases of z, some simplifications are performed.

Examples

Im(1+I)1

Im(1.2 - 3.4*I)-3.4

Im(-123)0

Im(Log(-6))Pi

Im(I*Infinity)Infinity

Simplifications are performed if possible.

Im(-2*x)-2*Im(x)

Im(3+Pi+z)Im(z)

Im(Abs(z))0

Im(Exp(2+3*I))E^2*Sin(3)

Indeterminateconstant

Syntax

Indeterminate

represents a mathematical quantity whose magnitude is undetermined.

Details

Calculations with at least one Indeterminate leads to an Indeterminate result.

Examples

0 / 0Indeterminate

0 ^ 0Indeterminate

Overflow * UnderflowIndeterminate

Infinity - InfinityIndeterminate

Infinityconstant

Syntax

Infinity

represents the mathematical positive infinite quantity.

Details

Infinity is the short form of DirectedInfinity(1).
Infinity is handled in calculations as an exact number and keeps exact computation possible, unlike Overflow.
Infinity is even larger than Overflow.

Examples

Log(0)-Infinity

Infinity > OverflowTrue

Arithmetical and mathematical functions are able to process Infinity:

1 / Infinity0

Exp(-Infinity)0

Tanh(Infinity)1

Exact evaluation is not possible with Overflow:

Tanh(Overflow)1.0

Log(0.0)Indeterminate

Integer`n`object

Syntax

n

with an arbitrary number of digits represents an integer number.

Integer

is the head of an integer number.

Details

Integer numbers can be of any size and are not limited to the word size of the processor.
Integer numbers are always exact values.

Examples

Head(123)Integer

12 + 3446

Working with large numbers:

32845723658234756 * 34659487645645654611384159533448787206618577180912776

11384159533448787206618577180912776 / 34659487645645654632845723658234756

3^200265613988875874769338781322035779626829233452653394495974574961739092490901302182994384699044001

InternalFormfunction

Syntax

InternalForm(exp)

gives the internal representation of exp without operator signs.

Details

InternalForm converts operators to the expression format with head and arguments.
InternalForm shows all structures usually hidden during output.

Examples

InternalForm(4+2*I)Complex(4, 2)

InternalForm(1/5)Rational(1, 5)

InternalForm(1+2*x)Plus(1, Times(2, x))

InternalForm(x?Integer)Pattern(x, Any(Integer))

Possible Issues

InternalForm visualizes the internal used expression structure, not the input structure.

InternalForm(a-b)Plus(a, Times(-1, b))

InternalForm(1/x)Power(x, -1)

InternalForm(Exp(z))Power(E, z)

Intersectionfunction

Syntax

Intersection(list₁, list₂, … )

gives a sorted list of all distinct elements that appear in all of list_i.

Details

If the lists list_i are considered as sets, Intersection gives their intersection list₁ ∩ list₂.
Intersection works with any kind of expressions as long as they have the same head.

Examples

Intersection( {4,3,3,1} , {1,4,4,5} ){1, 4}

Intersection( {x, y} , {u, v} ){}

Intersection( {1,2,3} , {1,10,100} , {1,2,4,8} ){1}

Intersection works with any expressions, not only lists, as long as they have the same head.

Intersection( a + b + d , a + c + d )a + d

Intersection( f(1, 4, 2), f(2, 3, 4) )f(2, 4)

Possible Issues

Heads have to be the same when Intersection is applied on expressions.

Intersection( f(1, 2), g(2, 3) )Intersection.Heads: Heads f and g are not the same.Intersection(f(1, 2), g(2, 3))

IsEvenfunction

Syntax

IsEven(n)

gives True if n is an integer and even, and False otherwise.

Details

IsEven has the attribute Listable.

Examples

IsEven(128)True

IsEven(-5)False

IsEven(4.0)False

IsEven(x)False

IsEven({1,2,3,4}){False, True, False, True}

IsInfinitefunction

Syntax

IsInfinite(z)

gives True if z is an infinite expression.

Details

IsInfinite gives True in case of Infinity, ComplexInfinity, and DirectedInfinity.
IsInfinite gives False in case of numerical overflows or underflows, like Underflow or ComplexOverflow.

Examples

IsInfinite(1/0)True

IsInfinite(Log(0))True

IsInfinite(I*Infinity)True

Expressions which are not infinite:

IsInfinite(123)False

IsInfinite(Exp(-Infinity))False

IsInfinite(Overflow)False

IsNumericfunction

Syntax

IsNumeric(exp)

gives True if exp is a numeric quantity, and False otherwise.

Details

IsNumeric(exp) gives True, when Calculate(exp) whould yield an explicit number.

Examples

Numeric expressions:

IsNumeric(3/7)True

IsNumeric(3+I)True

IsNumeric(Log(Sqrt(2)))True

IsNumeric(Pi)True

Non-numeric expressions:

IsNumeric(a+b)False

IsNumeric({1, 2, 3})False

Results in case of numerical boundaries:

IsNumeric(Underflow)True

IsNumeric(ComplexOverflow)True

IsNumeric(Infinity)False

IsOddfunction

Syntax

IsOdd(n)

gives True if n is an integer and odd, and False otherwise.

Details

IsOdd has the attribute Listable.

Examples

IsOdd(127)True

IsOdd(-4)False

IsOdd(3.0)False

IsOdd(x)False

IsOdd({1,2,3,4}){True, False, True, False}

IsOverflowfunction

Syntax

IsOverflow(z)

gives True if z is a numerical overflow.

Details

IsOverflow gives True in case of -Overflow, Overflow, and ComplexOverflow.
IsOverflow gives False in case of infinite expressions, like Infinity or ComplexInfinity.

Examples

IsOverflow(Overflow)True

IsOverflow(Exp(10.0^100))True

IsOverflow(ComplexOverflow)True

IsOverflow(1.0/Underflow)True

Expressions which have no overflow:

IsOverflow(123)False

IsOverflow(Exp(Infinity))False

IsOverflow(1/0)False

IsPrimefunction

Syntax

IsPrime(n)

gives True if n is prime, and False otherwise.

Details

IsPrime uses a deterministic variant of the Miller–Rabin primality test.
IsPrime becomes slow for n > 3×10²³.
IsPrime has the attribute Listable.

Examples

IsPrime(127)True

IsPrime(117)False

IsPrime(6549620699)True

IsPrime(79857529916387089718377)True

IsPrime({1,2,3,4,5}){False, True, True, False, True}

IsUnderflowfunction

Syntax

IsUnderflow(z)

gives True if z is a numerical underflow.

Details

IsUnderflow gives True in case of -Underflow, Underflow, and ComplexUnderflow.

Examples

IsUnderflow(Underflow)True

IsUnderflow(Exp(-10.0^30))True

IsUnderflow(ComplexUnderflow)True

IsUnderflow(1.0/Overflow)True

Expressions which have no underflow:

IsUnderflow(123)False

IsUnderflow(Exp(-Infinity))False

IsUnderflow(1/0)False

Iteratefunction

Syntax

Iterate(exp, n)

gives a list of n evaluated copies of exp.

Iterate(exp, {i, i_max} )

generates a list by iterating i from 1 to i_max.

Iterate(exp, {i, i_min, i_max} )

generates a list by iterating i from i_min to i_max.

Iterate(exp, {x, x_min, x_max, x_step} )

generates a list by iterating x from x_min to x_max using steps x_step.

Iterate(exp, {e, {e₁, e₂, …, e_n } } )

generates a list by iterating e though the values e₁, e₂, …, e_n.

Iterate(exp, spec₁, spec₂, … )

is evaluated as Iterate(Iterate(exp, spec₁), spec₂, … ).

Details and Options

The expression exp is just evaluated during the iteration, not before.
The iterator symbol (i, x or e) is used in a local scope and protected from outer scope.

Examples

Iterations without iterator:

Iterate(0, 8){0, 0, 0, 0, 0, 0, 0, 0}

Iterate("Text", 4){"Text", "Text", "Text", "Text"}

Iteration with an iterator:

Iterate(i, {i, 10} ){1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Iterate(i^2, {i, -3, 3} ){9, 4, 1, 0, 1, 4, 9}

Iterate(x, {x, 0.5, 0.9, 0.1} ){0.5, 0.6, 0.7, 0.8, 0.9}

Iterate(Sin(x), {x, 0, 4*Pi, Pi/2} ){0, 1, 0, -1, 0, 1, 0, -1, 0}

Iteration with an iterator and specific elements:

Iterate(w*2, {w, {1,0.7,t,1/3} } ){2, 1.4, 2*t, 2/3}

Nested iteration with two iterators m and n:

Iterate(m+n*10, {m, 1, 4}, {n, 1, 4}){{11, 12, 13, 14}, {21, 22, 23, 24}, {31, 32, 33, 34}, {41, 42, 43, 44}}

The outer iterator n can be used in the inner iteration range.

Iterate(m+n*10, {m, 1, n}, {n, 1, 4}){{11}, {21, 22}, {31, 32, 33}, {41, 42, 43, 44}}

Possible Issues

The expression to iterate is just evaluated during the iteration, not before. In the following, this leads to different random numbers instead of ten identically numbers.

Iterate(Random(9), 10){5, 1, 4, 3, 1, 2, 2, 1, 5, 3}

IterateParallelfunction

Syntax

IterateParallel(exp, n)

gives a list of n in parallel evaluated copies of exp.

IterateParallel(exp, {i, i_max} )

generates a list by iterating i from 1 to i_max.

IterateParallel(exp, {i, i_min, i_max} )

generates a list by iterating i from i_min to i_max.

IterateParallel(exp, {x, x_min, x_max, x_step} )

generates a list by iterating x from x_min to x_max using steps x_step.

IterateParallel(exp, {e, {e₁, e₂, …, e_n } } )

generates a list by iterating e though the values e₁, e₂, …, e_n.

IterateParallel(exp, spec₁, spec₂, … )

is evaluated as IterateParallel(IterateParallel(exp, spec₁), spec₂, … ).

Details and Options

IterateParallel is a parallel version of Iterate, using multiple kernels (processor cores) for evaluation.
In principle, the result on IterateParallel is identical to the result of Iterate, except for side effects during parallel computation.
During the first call of IterateParallel, the additional kernels will be started automatically.

Examples

Evaluations in parallel:

IterateParallel(x^2, {x, 1, 10} ){1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

IterateParallel(Fibonacci(i), {i, 1, 10} ){1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Evaluation time is reduced, depending on the available numbers of cores.

Timing(Iterate(Pause(1), 4);){4.00127, Null}

Timing(IterateParallel(Pause(1), 4);){1.00117, Null}

Evaluation time is reduced, but the result is identical.

Timing(Count(Iterate(IsPrime(n), {n, 2^70, 2^70+5000}), True)){0.213779, 116}

Timing(Count(IterateParallel(IsPrime(n), {n, 2^70, 2^70+5000}), True)){0.0450858, 116}

Possible Issues

Tivial computation may take longer in parallel than in serial.

Timing(Iterate(Sin(x), {x,0.0,1.0,0.01}); ){3.792e-4, Null}

Timing(IterateParallel(Sin(x), {x,0.0,1.0,0.01}); ){0.0021673, Null}

Variables are localized in there kernels regarding redefinitions. Here, each kernel increment his local version of x and the outer x keeps untouched:

QuitKernels(); LaunchKernels(3); x = 0; IterateParallel( x = x + 1; Print("Kernel "~ToString(KernelID)~", x = "~ToString(x)); , 10 );Kernel 13, x = 1Kernel 14, x = 1Kernel 15, x = 1Kernel 14, x = 2Kernel 13, x = 2Kernel 15, x = 2Kernel 14, x = 3Kernel 13, x = 3Kernel 15, x = 3Kernel 14, x = 4Null

x0

Join~function

Syntax

Join(exp₁, exp₂, … )exp₁ ~ exp₁ ~ …

joins all expressions exp_i that have the same head, e.g. List{…}.

Join(str₁, str₂, … )str₁ ~ str₁ ~ …

joins all strings str_i.

Examples

Join( {1,2} , {3,4,5} ){1, 2, 3, 4, 5}

Join( "Hello ", "World!" )"Hello World!"

Join works with any expressions, not only lists, as long as they have the same head.

Join( a + b , c + d )a + b + c + d

Join( f(1, 2), f(x) )f(1, 2, x)

KernelIDconstant

Syntax

KernelID

gives the unique ID of the processing kernel.

Details

KernelID is 0 for the master kernel.
KernelID is positive unique integer for each parallel launched kernel.

Examples

KernelID0

Iterate(KernelID, 10) IterateParallel(KernelID, 10){0, 0, 0, 0, 0, 0, 0, 0, 0, 0} {1, 2, 3, 4, 1, 2, 3, 1, 2, 3}

Kernelsfunction

Syntax

Kernels()

gives a List{…} of IDs of all parallel launched kernels.

Details

Kernels can be used to receive the KernelID of all currently running kernels used for parallel computation.
LaunchKernels and QuitKernels can be used to launch and quit kernels, respectively.

Examples

Perform any parallel evalutation.

IterateParallel(i^2, {i, 0, 16}){0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256}

Receive the IDs of all parallel launched kernels:

Kernels(){1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

LaunchKernelsfunction

Syntax

LaunchKernels()

launches kernels for parallel computation depending on the number of processor cores.

LaunchKernels(n)

launches n kernels for parallel computation.

Details

LaunchKernels can be used to launch a specific number of kernels for parallel computing.
LaunchKernels gives a List{…} of IDs of parallel launched kernels.
It is recommended to keep n smaller than the available number of processor cores.
Functions like IterateParallel or MapParallel automatically launch kernels for parallel evalutation.
QuitKernels can be used to quit parallel launched kernels.

Examples

Launch four kernels for subsequent parallel computing:

LaunchKernels(4){1, 2, 3, 4}

IterateParallel(KernelID, 10){1, 2, 3, 4, 1, 2, 3, 1, 2, 3}

An other call of LaunchKernels launches more additional kernels, which can be checked with Kernels:

LaunchKernels(3){5, 6, 7}

Kernels(){1, 2, 3, 4, 5, 6, 7}

Lengthfunction

Syntax

Length(exp)

gives the number of elements in expression exp.

Length(str)

gives the number of characters in string str.

Details

For neither expression nor string, Length is 0.

Examples

Length( {1,2,3,4,5} )5

Length( "Hello World!" )12

Length counts the stored characters, not the metacharacters.

Length( '\0\x80ÿ\n' )4

Length works with any expressions, not only lists or strings.

Length( a + 2*b + c )3

Length( f(a, b) )2

Rational and complex numbers have a length of 0.

Length( 1/3 )0

Length( 1+2*I )0

Possible Issues

Length operates on the internal evaluated form, not on the displayed form.

Minus(x) Length( -x )-x 2

InternalForm( -x )Times(-1, x)

Less<function

Syntax

Less(x₁, x₂, … )x₁ < x₂ < …

gives True if x_i is less than x_i + 1 for all arguments, and False if x_i is greater than or equal to x_i + 1 for at least one pair.

Details

For exact expressions in x_i, Less compares the approximated value.

Examples

1 < 2True

-0.5 < 0 < 1/3True

x < 2 < 2False

The comparison will be simplified if possible.

1 < 2 < x2 < x

The value of an exact expression is approximated for comparison.

Sin(4) < 0True

LessEqual<=function

Syntax

LessEqual(x₁, x₂, … )x₁ <= x₂ <= …

gives True if x_i is less than or equal to x_i + 1 for all arguments, and False if x_i is greater than x_i + 1 for at least one pair.

Details

For exact expressions in x_i, LessEqual compares the approximated value.

Examples

1 <= 2True

-0.5 <= 1.5 <= 3/2True

x <= 2 <= 1False

The comparison will be simplified if possible.

1 <= 2 <= x2 <= x

The value of an exact expression is approximated for comparison.

Sin(4) <= 0True

List{…}function

Syntax

List(e₁, e₂, … ){ e₁, e₂, … }

represents a list of elements e_i.

Details

Lists are not exceptional constructs, but just an expression with the head List.
Most built-in functions have the attribute Listable and thread into lists.

Examples

List( 1, 2, 3 ){1, 2, 3}

{{a1,a2}, {b1,b2}}{{a1, a2}, {b1, b2}}

Most built-in functions thread into lists:

Sqrt({1, 4, 9}){1, 2, 3}

Exp({0.0, 1.0, 10.0}){1.0, 2.71828, 22026.5}

Listableattribute

Syntax

Listable

is an attribute for symbols to indicate that the symbol will threaded into lists occurring in arguments.

Details

Typical functions like Plus+ and Times* have the attribute Listable.
Attributes can be set and get with Attributes.

Examples

{a, b} + {c, d}{a + c, b + d}

Sin({x, 2, 0.5}){Sin(x), Sin(2), 0.479426}

By default the head of an expression is not threaded into the list.

f({1, 2, 3})f({1, 2, 3})

After setting the attribute Listable, f will threaded into the list.

Attributes(f) = Listable{Listable}

f({1, 2, 3}){f(1), f(2), f(3)}

f({1, 2, 3}, {a, b, c}){f(1, a), f(2, b), f(3, c)}

Lockedattribute

Syntax

Locked

is an attribute for symbols to indicate that they are locked against changes in their attributes.

Details

Some fundamental built-in symbols and functions are locked.
Attributes can be set and get with Attributes.

Examples

Attributes(Plus) = FlatAttributes.Locked: Plus is locked!{}

After setting the attribute Locked, no more changes in attributes can be made.

Attributes(f) = {Flat, Locked}{Flat, Locked}

Attributes(f) = {Orderless}Attributes.Locked: f is locked!{}

Attributes(f){Flat, Locked}

Logfunction

Function Plot

Syntax

Log(z)

gives the natural logarithm of z.

Log(b, z)

gives the logarithm of z to base b.

Details

Log gives an exact result when possible.
If z is an exact numerical value, Log remains unevaluated if no exact solution was found.
Log(b, z) is converted to Log(z)/Log(b) when no exact solution was found.
Log has the attribute Listable.
For calculations with an arbitrary precision a series based on the area hyperbolic tangent function is used.

Examples

Log(1000.0)6.90776

Log(-3.14)1.14422+3.14159*I

Log(3.0+2.0*I)1.28247+0.588003*I

Calculate(Log(100), 70)4.605170185988091368035982909368728415202202977257545952066655801935145

Particular arguments and their result:

Log(E)1

Log(0)-Infinity

Log(-1)I*Pi

Logarithm with specified base:

Log(10, 1000)3

Log(4.5, 4.5^100000)1.0e5

Log(2/3, 16/81)4

Log(1/4, 8)-3/2

Possible Issues

If an exact solution does not exist, just a conversion is performed.

Log(2, 100)Log(100)/Log(2)

MachinePrecisionconstant

Syntax

MachinePrecision

indicates the usage of machine precision in float pointing numbers like reals.

Details

MachinePrecision is exact and is not evaluated unless it is combined with an inexact computation.
For 64-bit IEEE float pointing numbers MachinePrecision is 53/Log(2,10).
Precision gives MachinePrecision if the number is stored as a machine number.

Examples

Calculate MachinePrecision on current system:

Calculate(MachinePrecision)15.9546

Small real numbers with low precision stored with machine precision:

Precision(3.1415)MachinePrecision

Precision(789.456e123)MachinePrecision

Large real numbers stored with an arbitrary precision with at least machine precision:

Precision(123.4e5678)15.9546

Map/@function

Syntax

Map(f, exp)f /@ exp

applies f to each element in expression exp.

Details

After f is applied, the whole expression is evaluated.

Examples

Map(f, g(1, a, x))g(f(1), f(a), f(x))

Map(Cos, Pi+b+c)-1 + Cos(b) + Cos(c)

Map will process anonymous function as well:

#^2 & /@ {4, 9, a, 0.4}{16, 81, a^2, 0.16}

MapParallelfunction

Syntax

MapParallel(f, exp)

applies f in parallel to each element in expression exp.

Details

MapParallel is a parallel computing version of Map/@, using multiple kernels (processor cores) for evaluation.
In principle, the result of MapParallel is identical to the result of Map/@, except for side effects during parallel computation.
During the first call of MapParallel, the additional kernels will be started automatically.
After f is applied, the whole expression is evaluated in series.

Examples

MapParallel(f, {1, a, x}){f(1), f(a), f(x)}

MapParallel(Cos, {Pi, 1.0, I, x}){-1, 0.540302, Cosh(1), Cos(x)}

Evaluation time is reduced, depending on the available numbers of cores.

Timing( Map(Pause, {0.5, 1.0, 1.5, 2.0} ) ){5.00082, {Null, Null, Null, Null}}

Timing( MapParallel(Pause, {0.5, 1.0, 1.5, 2.0} ) ){2.00053, {Null, Null, Null, Null}}

Evaluation time is reduced, but the result is identical.

testlist = Random(2^70, 5000); /* Create a list of 5000 random huge numbers. */Null

Timing( Count(Map(IsPrime, testlist), True) ){0.267339, 107}

Timing( Count(MapParallel(IsPrime, testlist), True) ){0.0458072, 107}

Matchingfunction

Syntax

Matching(exp, pat)

performs a pattern matching of pattern pat on expression exp.

Details

Matching gives True if pat matches exp, and False otherwise.
pat usually consists of patterns like Any?, AnySequence??, or Alternatives|, or Condition/?.

Examples

Matching(123, ?Integer) Matching(123.0, ?Integer)True False

Matching({a, b, c}, {??Symbol}) Matching({a, 2, b}, {??Symbol})True False

Matching({10, 11}, {x?, x?}) Matching({10, 10}, {x?, x?})False True

Matching({2, 1}, {x?, y?} /? x>y ) Matching({1, 2}, {x?, y?} /? x>y )True False

Maxfunction

Syntax

Max(x₁, x₂, … )Max( {x₁, x₂, … }, {x_n + 1, x_n + 2, … }, … )

gives the numerically largest value of x_i.

Details

Max gives only in case of exclusively real numerical values x_i the largest value.
For exact expressions in x_i, Max compares the approximated value.
Max() gives -Infinity.
Max(exp) gives exp.

Examples

Max(3, -2, 1.0)3

Max(E, Pi)Pi

Max({4,2,-4}, {9, 0, 4})9

Possible Issues

Max remains unevaluated, but simplified, when non-comparable values are included.

Max(3, "test", -5, 2+I, 4)Max(4, "test", 2+I)

Minfunction

Syntax

Min(x₁, x₂, … )Min( {x₁, x₂, … }, {x_n + 1, x_n + 2, … }, … )

gives the numerically smallest value of x_i.

Details

Min gives only in case of exclusively real numerical values x_i the smallest value.
For exact expressions in x_i, Min compares the approximated value.
Min() gives Infinity.
Min(exp) gives exp.

Examples

Min(3, -2, 1.0)-2

Min(E, Pi)E

Min({4,2,-4}, {9, 0, 4})-4

Possible Issues

Min remains unevaluated, but simplified, when non-comparable values are included.

Min(3, "test", -5, 2+I, 4)Min(-5, "test", 2+I)

Minus-function

Syntax

Minus(exp)-exp

gives the arithmetic negation of exp.

Details

Minus(exp) is converted to Times(-1, exp).
Minus has the attribute Listable.

Examples

Minus(123)-123

Minus(a+b+c)-a - b - c

Minus(3+2*I)-3-2*I

Double negation of an expression is the expression itself.

Minus(Minus(exp))exp

Modfunction

Syntax

Mod(m, n)

gives the remainder on devision of m by n.

Details

The sign of n is used as the sign of the remainder.
Mod is suitable for both symbolic and numerical manipulation.
Mod works also with rational, real, and complex numbers.
Mod has the attribute Listable.
In cases of general expressions in m and n simplifications are tried.

Examples

Mod(7, 3)1

Mod(4.0, 1.2)0.4

Mod(5/9, 1/2)1/18

Mod(7+8*I, 3+3*I)1+2*I

Mod also performs on expressions.

Mod(Pi, 2)-2 + Pi

Mod(8*Pi, 3*Pi)2*Pi

Mod(10, 5^(1/2))10 - 4*5^(1/2)

Mod(14*a^2, 3*a^2)2*a^2

Possible Issues

The result of Mod(m, n) is always in the range 0 to n, and negative when n is negative.
This behavior differs from other languages, where the result becomes negative when m is negative, instead of n.

Mod(7, 3)1

Mod(-7, 3)2

Mod(7, -3)-2

Mod(-7, -3)-1

Nestfunction

Syntax

Nest(f, exp, n)

gives an expression with f applied n times to exp.

Details and Options

Usually, f is a symbol or an anonymous function.

Examples

Nest(f, x, 4)f(f(f(f(x))))

Nest(Function((#+1)^2), x, 3)(1 + (1 + (1 + x)^2)^2)^2

Searching for fix-points:

Nest(Cos, 0.0, 100)0.739085

Cos(0.739085)0.739085

Imitate exponential growth:

Nest(Function(1.02*#), 100.0, 10)121.899

100.0 * 1.02^10121.899

Not!function

Syntax

Not(bool)!bool

gives the logical NOT of bool.

Not(int)!int

gives the binary NOT of the integer int.

Not(str)!str

gives the binary NOT of the string str.

Details

Not gives True if bool is False, and False if it is True.
Integers are handled in their two's complement form.
Strings are converted into binary form and Not is performed bitwise.
Simplifications are performed if possible to evaluate Not on expressions.

Examples

! FalseTrue

! 15-16

! 'BINARY''½¶±¾¦'

Negation of comparisons are simplified if possible.

Not( x < 3 )x >= 3

Not( x == 3 )x != 3

Double negation of an expression is the expression itself.

Not( Not( exp ) )exp

Nullconstant

Syntax

Null

represents the absence of any kind of object.

Details

Null is usually returned from Compound;, Print or delayed assignments like DefineDelayed:= or RuleDelayed:>.
A single Null will be hidden from the output.

Examples

NullNull

x := 1Null

z = 3;Null

{1, Print(2), z}2{1, 2, 3}

Numeratorfunction

Syntax

Numerator(z)

gives the numerator of the number z.

Details

For a rational number z = a/b, Numerator gives a.
For integer and real numbers Numerator(z) = z.
Numerator picks out terms with a non-negative exponent.
Numerator has the attribute Listable.

Examples

Numerator(3/7)3

Numerator(-8/3)-8

Numerator(123)123

Numerator(3.4)3.4

Numerator(1/2 + 1/3*I)3+2*I

Numerator(f/g)f

Numerator(a^2*b^(-3)*c^4)a^2*c^4

Optional:function

Syntax

Optional(pat, exp )pat : exp

is a pattern object that is replaced by exp if pat is not matched.

Details

Optional can be used to define a default value in function's arguments.
Optional can be used at any position, not only at the end of arguments.

Examples

Matching( {1, 2}, {x?, y?:0} ) Matching( {1}, {x?, y?:0} )True True

f(x?, y?:2) := x^yNull

f(a, b) f(a)a^b a^2

g(n?Integer:0, str?String) := {n, str}Null

g("Text") g(10, "Text"){0, "Text"} {10, "Text"}

Optionsfunction

Syntax

Options(sym₁ -> val₁, sym₂ -> val₂, … )Options(sym₁ :> val₁, sym₂ :> val₂, … )

is a pattern object that matches a collection of rules for symbols sym_n and their default value val_n.

Details

Options is typically used in definition of functions for additional arguments attributed by their names.
Arguments and default values in Options can either by defined with Rule-> or RuleDelayed:>.

Examples

f(Options(x->1, y->2)) := {x, y}Null

f() f(x->100) f(y->100) f(y->100, x->99){1, 2} {100, 2} {1, 100} {99, 100}

Arguments defined with RuleDelayed are evaluated each time again they will be used:

f(Options(x->0)) := {x, x, x, x, x}Null

f() f(x->Random(10)) f(x:>Random(10)){0, 0, 0, 0, 0} {1, 1, 1, 1, 1} {0, 6, 3, 7, 1}

Or||function

Syntax

Or(bool₁, bool₂, … )bool₁ || bool₂ || …

gives the logical OR of bool_i.

Or(int₁, int₂, … )int₁ || int₂ || …

gives the binary OR of the integers int_i.

Or(str₁, str₂, … )str₁ || str₂ || …

gives the binary OR of the strings str_i.

Details

The arguments of Or are evaluated step by step.
Or gives True immediately if any argument is True, and False if all arguments are False.
Or() is defined as False.
Or(exp) is simplified to exp.
Integers are handled in their two's complement form.
Strings are converted into binary form and Or is performed bitwise.

Examples

True || FalseTrue

12 || 715

'BINARY' || 'abc''ckoARY'

Or stops to evaluate its arguments if True occurs. In the following case the second Print is not evaluated.

(Print(1);False) || True || (Print(2);False)1True

Expressions which are evaluated into False will be erased.

1==2 || a==5 || b==10 || Falsea == 5 || b == 10

Orderfunction

Syntax

Order(exp₁, exp₂)

gives 1 if exp₁ and exp₂ are in order, gives -1 if exp₁ and exp₂ are out of order, and gives 0 if both expression are identical.

Details

Order is used as default comparison function in Sort.

Order gives the typical order of exp₁ and exp₂ in orderless expressions:

Integern, Rationala/b and Realx numbers are ordered by their numerical value.

Complexx+y*I numbers are ordered by their real part and then by their imaginary part.

String"…",'…' are ordered using the Unicode collation algorithm.

Numerical values are ordered before String"…",'…', Symbols and Expressionf(…).

Examples

Order(123, var)1

Order(1, 1+I)1

Order("b", "a")-1

Order(x^2, x^2)0

Possible Issues

Constants are ordered by their name and not by their numerical value.

Order(Pi, 4)-1

Orderlessattribute

Syntax

Orderless

is an attribute for symbols to indicate that their arguments can be sorted and rearranged in pattern matching.

Details

Typical functions like Plus+ and Times* have the attribute Orderless.
Attributes can be set and get with Attributes.

Examples

Summands in Plus+ are sorted during evaluation:

a + c + ba + b + c

By default expressions keep their order of arguments.

f(1, b, 2, a)f(1, b, 2, a)

Matching( f(x, 2) , f(?Integer, x) )False

After setting the attribute Orderless, expressions will sorted and pattern matching tries all combinations.

Attributes(f) = Orderless{Orderless}

f(1, b, 2, a)f(1, 2, a, b)

Matching( f(x, 2) , f(?Integer, x) )True

Outputfunction

Syntax

Output(exp₁, exp₂, … )


exp₁
exp₂
 …

evaluates all expressions exp_i in turn.

Details

Output generates a newline (carriage return and line feed) only on top level.
In in deeper levels Output remains to avoid syntax errors.

Examples

Output(1+2, 3*4, 5^6)3 12 15625

1+2 3*4 5^63 12 15625

f(Output(1,2,3))f(Output(1, 2, 3))

Overflowconstant

Syntax

Overflow

represents an overflow in float-pointing or integer arithmetic.

Details

The head of Overflow is Real.
Overflow is always larger than the highest possible real value, but always smaller than Infinity.

Examples

Exp(10.0^20.0)Overflow

Infinity > OverflowTrue

Arithmetical and mathematical functions are able to process Overflow:

3.14 + OverflowOverflow

Overflow * (-2)-Overflow

3.14 / OverflowUnderflow

ArcTan(Overflow)1.5708

Part[…]function

Syntax

Part(exp, level₁, level₂, … )exp[level₁, level₂, … ]

gives a part of the expression exp specified by level_i for the different nesting levels.

Part(str, part)str[part]

gives a part of the string str specified by part.

Details

The part and level specifications can have the following forms:

`…[n]`	gives the `n`^th element of an expression or the `n`^th character of a string.
`…[-n]`	counts from the end of the expression or the string.
`…[0]`	gives the head of an expression or the string.
`…[{n₁, n₂, … }]`	gives the expression with elements specified by `n_i` or the string with characters specified by `n_i`.
`…[m..n]`	gives the expression with a span of elements or the string with characters from `m` until `n`.
`…[m..n..s]`	gives a span of elements or characters from `m` until `n` in steps of `s`, where `s` can also be negative.
`…[All]`	gives the expression with all elements in this level or the string with all characters.

Examples

Pick an element or a character:

Part( {a,b,c,d,e} , 3 )c

"Hello World!"[-1]"!"

Pick multiple elements or characters:

{a,b,c,d,e}[{-1,3,3}]{e, c, c}

"Hello World!"[{1,8,3,-2}]"Hold"

Pick a span of elements or characters:

{a,b,c,d,e}[2..-2]{b, c, d}

{a,b,c,d,e}[-1..1..-1]{e, d, c, b, a}

Part( "Hello World!" , 1..7..2 )"HloW"

Pick all 2^nd elements from the second level:

{{a1,a2}, {b1,b2}, {c1,c2}}[All, 2]{a2, b2, c2}

Part works on any expression.

Part( a+b*x+x^2 , 2, 1)b

f(1, 2, x, 4, y)[3..5]f(x, 4, y)

Possible Issues

Rationala/b numbers or Complexx+y*I numbers are atomic objects. Part cannot extract the numerator/denominator or real/imaginary part. In such cases you have to use Numerator, Denominator, Re, or Im.

Part( 3/4 , 1 ) Numerator( 3/4 )Part.Depth: Part specification (3/4)[1] is longer than depth of object.(3/4)[1] 3

Part( 2+3*I , 1 ) Re( 2+3*I )Part.Depth: Part specification (2+3*I)[1] is longer than depth of object.(2+3*I)[1] 2

Expressions like Plus+ and Times* are ordered and flattened before Part is executed.

Part( a+c+b , 2) a+c+b // InternalFormb Plus(a, b, c)

Part( (a*b)*(c*d) , 2) (a*b)*(c*d) // InternalFormb Times(a, b, c, d)

Partitionfunction

Syntax

Partition(str, n)

partitions the string str into a list of sub-strings of length n.

Partition(exp, n)

partitions the expression exp into sub-expressions of length n.

Details

If the length of exp is no multiple of n, some elements will be disregarded.

Examples

Partition( {1, 2, 3, 4, 5, 6} , 3 ){{1, 2, 3}, {4, 5, 6}}

Partition( {1, 2, 3, 4, 5, 6} , 4 ){{1, 2, 3, 4}}

Partition( "Hello World!" , 3 ){"Hel", "lo ", "Wor", "ld!"}

Partition( "Hello World!" , 5 ){"Hello", " Worl"}

Partition( f(1,2,3,4) , 2 )f(f(1, 2), f(3, 4))

Patternfunction

Syntax

Pattern(sym, pat)

represents the pattern pat and assigns a matching object to the symbol sym.

sym?
sym??
sym???

is the short form for Pattern(sym, ?), Pattern(sym, ??) or Pattern(sym, ???).

sym?head
sym??head
sym???head

is the short form for Pattern(sym, ?head), Pattern(sym, ??head) or Pattern(sym, ???head).

Details

Pattern is a common argument for function definitions with DefineDelayed:= or SubDefineDelayed@:=.
Pattern can be used in RuleDelayed:> within Replace/:.

Examples

Using the short form of Pattern in functional definitions:

f(x?, y?) := x ^ y f(a, 2)a^2

g(x??) := Times(x) g(3) g(4, 5) g(a, 6, x^2)3 20 6*a*x^2

h(Pattern(x, ?Integer|Infinity)) := x^2 { h(1), h(-5), h(Infinity), h(3.14) }{1, 25, Infinity, h(3.14)}

Using Pattern during a replacement:

Replace( {2, 3, x} , x? :> x^2 ){4, 9, x^2}

Replace( {1, 2, a, b, c, 3.14} , Pattern(x, a|b) :> x^3 ){1, 2, a^3, b^3, c, 3.14}

Pausefunction

Syntax

Pause(x)

pauses the evaluation for at least x seconds.

Details

x is a positive integer, rational or real number.

Examples

Timing(Pause(2)){2.00004, Null}

Timing(Pause(0.3)){0.300035, Null}

Piconstant

Syntax

Pi

represents the constant π ≅ 3.14159.

Details

Pi is exact and is not evaluated unless it is combined with an inexact computation.
Pi can be calculated with arbitrary precision.
For calculations with an arbitrary precision the Bailey–Borwein–Plouffe formula is used.

Examples

Calculate Pi:

Calculate(Pi)3.14159

Calculate(Pi, 70)3.141592653589793238462643383279502884197169399375105820974944592307816

Arithmetic with Pi:

2*Pi + 3*Pi5*Pi

2.0*Pi6.28319

Some functions give exact results for terms with Pi.

Sin(Pi)0

Expressions with Pi are returned by some functions.

ArcCos(-1)Pi

ArcTan(Infinity)1/2*Pi

Log(-1)I*Pi

References

Bailey D H, Borwein P B, Plouffe S (1997), "On the Rapid Computation of Various Polylogarithmic Constants", Mathematics of Computation 66 (218): 903–913

Plus+function

Syntax

Plus(exp₁, exp₂, … )exp₁ + exp₂ + …

gives the sum of all expressions exp_i.

Details

Plus is suitable for both symbolic and numerical manipulation.
Plus has the attributes Orderless, Flat and Listable.
Plus() is defined as 0.
Plus(exp) is simplified to exp.

Examples

1 + 2 + 36

2*x + 3*x5*x

2 + 1/4 2 + 0.259/4 2.25

Plus is flattened and the arguments are sorted.

(a + c) + (b + 4)4 + a + b + c

Plus threads over List{…}:

{1, 2, 3} + 10{11, 12, 13}

{1, 2, x} + {5, 10, y}{6, 12, x + y}

Power^function

Syntax

Power(exp₁, exp₂)exp₁ ^ exp₂

gives expression exp₁ to the power of exp₂.

Details

Power is suitable for both symbolic and numerical manipulation.
Power has the attribute Listable.
In case of exact arguments, Power tries to evaluate an exact number.
For complex numbers and calculations with an arbitrary precision the identity x^y = Exp(y*Log(x)) is used.

Examples

2 ^ 1001267650600228229401496703205376

5 ^ (-2)1/25

27 ^ (1/3) 10 ^ (1/3)3 10^(1/3)

10 ^ (1/3) // Calculate 10.0 ^ (1/3)2.15443 2.15443

2.5 ^ 10008.709809816217e397

Power threads over List{…}:

{2, 3, 4} ^ 2{4, 9, 16}

{2, 10} ^ {4, 3}{16, 1000}

Possible Issues

Some evaluations give Indeterminate or ComplexInfinity:

0^0Indeterminate

0^(-1)ComplexInfinity

If the result of Power is to large or near zero, it gives Overflow or Underflow.

10.0 ^ (10.0^20.0)Overflow

0.5 ^ (10.0^20.0)Underflow

Precisionfunction

Syntax

Precision(z)

gives the number of precise decimal digits of the number z.

Details

In case of integers and rational numbers, Precision gives Infinity.
In case of real numbers, Precision gives MachinePrecision if the number is stored as a 64-bit IEEE float pointing number.
In case of complex numbers, Precision gives lowest precision of real part and imaginary part.
In case of an expression, Precision gives lowest precision of all elements.

Examples

Determine the number's precision:

Precision(123)Infinity

Precision(3/7)Infinity

Precision(3.14)MachinePrecision

Precision(Overflow)0.0

Precision(123456.78901234567890)20.0915

Precision(4.0 + 3*I)MachinePrecision

Printfunction

Syntax

Print(exp₁, exp₂, …, exp_last )

evaluates and prints all expressions exp_i in turn, and gives the last one exp_last as result.

Details

If exp_n is a string, only its content is shown and control characters are taken into account.
If exp_i is Null no print is performed. Using Null in exp_last, the result is hidden from output.
Print can be used to trace calculations in a more complex evaluation.

Examples

Print(1, 2, 3)1233

Print("String are shown\r\nwith line breaks\tand tabs.")String are shown with line breaks and tabs."String are shown\r\nwith line breaks\tand tabs."

Print(Hello, World, Null)HelloWorldNull

Trace a large calculation:

Print(Log(10, Print(2^10)-24 // Print)) * 2110241000363

Apply(Plus, Iterate(Print(Random(100)), 5))559556129209

Protectedattribute

Syntax

Protected

is an attribute for symbols to indicate that they are protected against changes by assignments or Clear.

Details

All built-in symbols and functions are protected.
Attributes can be set and get with Attributes.

Examples

Pi = 3Define.Protected: Pi is protected!3

After setting the attribute Protected, no more definitions can be made.

x = 123123

Attributes(x) = Protected{Protected}

x = 0Define.Protected: x is protected!0

x123

QuitKernelsfunction

Syntax

QuitKernels()

quits all kernels used for parallel computation.

QuitKernels(n)

quits n kernels used for parallel computation.

QuitKernels({id₁, id₂, … })

quits kernels with the specified IDs id_i.

Details

QuitKernels can be used to force the quit of a specific number of parallel launched kernels.
QuitKernels gives a List{…} of IDs of quited kernels.

Examples

First, launch 8 kernels to quit some of them later:

LaunchKernels(8){1, 2, 3, 4, 5, 6, 7, 8}

Quit the last three kernels:

QuitKernels(3){6, 7, 8}

Kernels(){1, 2, 3, 4, 5}

Quit two kernels with the spezified IDs:

QuitKernels({2, 5}){2, 5}

Kernels(){1, 3, 4}

Quit all kernels:

QuitKernels(){1, 3, 4}

Kernels(){}

Randomfunction

Syntax

Random(z)

gives a pseudo-random number between 0 and z.

Random(z, n)

gives a list of n pseudo-random numbers between 0 and z.

Random(str)

gives a pseudo-random character from the string str.

Random(str, n)

gives a string of length n with pseudo-random characters from the string str.

Random(list)

gives a pseudo-random element from the list list.

Random(list, n)

gives a list of n pseudo-random element from the list list.

Details and Options

Depending on the head of z (Integer, Rational, Real or Complex), Random gives either only integer numbers, rational numbers, real numbers, or complex numbers, respectively.
For a complex number z, Random is performed for the real part and imaginary part.
For a rational number z = a/b, the resulting numbers are in steps of 1/b.
By using an element in list or a character in str multiple times, this element/character can be weighted higher in the result.
Random('', n) gives a pseudo-random binary string of length n.

Examples

Single random numbers, characters or elements:

Random(7)6

Random(30!)127744442400524630741569973393108

Random(2/3)0

Random(2.0)1.46042

Random(1.0+2.0*I)0.303707+1.77864*I

Random("abcdefgh")"d"

Random( {a, b, c} )c

List of random numbers or elements:

Random( 9, 10 ){0, 6, 5, 4, 3, 3, 3, 2, 9, 8}

Random( 1.0, 10 ){0.435244, 0.0371735, 0.673563, 0.323152, 0.760165, 0.665368, 0.704443, 0.584308, 0.939667, 0.242821}

Random( {True, False}, 10 ){False, True, True, False, True, False, True, True, True, False}

Generate a string with random characters:

Random("ABC", 20)"CCBBBCACCABACAACCCAA"

Random("---|", 40)"---------||----|--|-||-----|-|------|-|-"

Generate a random binary string:

Random('', 32)'¼øÚ÷àï\x80n÷\x7F«-\x89´\x11%4\x85«GÕr,NÇrj\x9C\x81BxÆ'

Rational`a`/`b`object

Syntax

Rational(a, b)a/b

is a rational number, represented as a fraction of two integer numbers with numerator a and denominator b.

Rational

is the head of a rational number.

Details

The numerator a and denoninator b can be numbers of any size and are not limited to the word size of the processor.
Rational numbers are reduced as much as possible.
Rational numbers are always exact values.
a/b uses the same operator level as Divide/.

Examples

3/73/7

Rational(3, 7)3/7

Head(3/7)Rational

5 + 1/316/3

1/123456 + 1/678901802357/83814401856

Rational number are reduced as much as possible.

4/121/3

6/23

Many functions can handle and return rational numbers:

(2/3)^1001267650600228229401496703205376/515377520732011331036461129765621272702107522001

ArcSin(1/2)1/6*Pi

Log(2, 1/8)-3

Bernoulli(10)5/66

Rationalizeoption

Syntax

Rationalize -> FalseRationalize -> True

is an option to define whether integers are converted to rationals when shifting bits.

Details

Rationalize can be used in ShiftLeft and ShiftRight.
If Rationalize is not explicitly specified, False is used, meaning no rationalization is performed.

Examples

ShiftRight(15) ShiftRight(15, Rationalize->False) ShiftRight(15, Rationalize->True)7 7 15/2

ShiftRight(4, 5) ShiftRight(4, 5, Rationalize->True)0 1/8

Refunction

Syntax

Re(z)

gives the real part of the complex number z.

Details

If z is a real number, Re(z) is just z.
Re has the attribute Listable.
For special cases of z, some simplifications are performed.

Examples

Re(1+I)1

Re(1.2-3.4*I)1.2

Re(-123)-123

Re(Log(-6*I))Log(6)

Re(ArcCos(2))0

Re(Infinity)Infinity

Simplifications are performed if possible.

Re(-2*x)-2*Re(x)

Re(3+Pi+z)3 + Pi + Re(z)

Re(Abs(z))Abs(z)

Re(Exp(2+3*I))E^2*Cos(3)

Real`x`object

Syntax

x

with an arbitrary number of digits and a decimal point represents a real number.

Real

is the head of a real number.

Details

Real numbers can be of any precision and magnitude. Actually, the largest possible number is around 2^2⁵² ≈ 10^{1355718576299647}
Real numbers are stored with machine precision (64-bit IEEE float pointing number) if possible.
Real numbers with machine precision displayed only with 6 precision digits, but full IEEE precision is tracked.
Real numbers are inexact values and the precision can be received with Precision.
Calculations with at least one inexact value lead to an inexact result.
Overflow and Underflow are two special real numbers representing a float pointing overflow or underflow.

Examples

Head(1.0)Real

4.0 * 0.72.8

1 + 2.03.0

1/3 + 1.01.33333

1.0000000000000000000000000000000000000000123 + 1011.0000000000000000000000000000000000000000123

The magnitude of real numbers can be larger then typical 64-bit IEEE float pointing numbers can handle.

123.456 ^ 1000.03.25238259113e2091

0.012 ^ 1000.01.517910089172e-1921

Small real numbers with low precision are stored always with machine precision:

Precision(123.456)MachinePrecision

Precision(789.456e123)MachinePrecision

The precision of larger real numbers or numbers with higher precision is tracked:

x = 123.456 ^ 1000.0 y = 123.000000000000000000004567893.25238259113e2091 123.00000000000000000000456789

{ Precision(x) , Precision(y) }{12.4798, 29.0899}

x ^ y1.0018442638e257256

Precision(x^y)10.3898

Overflow and Underflow are real numbers.

123.456 ^ (10^100)Overflow

Head(Overflow)Real

123.456 ^ (-10^100)Underflow

Head(Underflow)Real

Possible Issues

Combinations of Overflow and Underflow can result in Indeterminate or complex numbers.

Overflow * UnderflowIndeterminate

Underflow - UnderflowComplexUnderflow

Replace/:function

Syntax

Replace(exp, rule)exp /: rule

applies the rule rule to the expression exp and its sub-expressions as a substitution.

Replace(exp, {r₁, r₂, … })exp /: {r₁, r₂, … }

applies the rules r_i to the expression exp and its sub-expressions as a substitution.

Replace(exp, rule, depth)

applies the rule rule to the expression exp and its sub-expressions down to depth level depth.

Replace(str, a->b )str /: a->b

replaces all sub-strings a in the string str with b.

Replace(str, {a₁->b₁, a₂->b₂, … } )str /: {a₁->b₁, a₂->b₂, … }

replaces all sub-strings a_i in the string str with b_i.

Details

Replace applies the rules starting from the lowest depth to the deepest level.
Without specification of depth or if depth = Infinity, replacements are performed as deep as possible.
Rules can be defined as Rule-> as well as RuleDelayed:>.
Once a rule was applied to a sub-expression, no more replacements will be applied to this sub-expression.
Once a rule was applied to a string part, no more replacements will be applied to this part.
When one of the strings str, a_i or b_i is binary, the result is binary too.

Examples

Replace a symbol with a value:

x^2 + x /: x->420

{x+y, x*y, x^y} /: {x->2, y->3}{5, 6, 8}

Swapping elements:

x^2 /: {2->x, x->2}2^x

Replace, with depth level specification:

{x, x, {x, x, {x, x}}, x} /: x->0{0, 0, {0, 0, {0, 0}}, 0}

Replace( {x, x, {x, x, {x, x}}, x} , x->0 , 2 ){0, 0, {0, 0, {x, x}}, 0}

Replace uses the Flat and Orderless attributes to find a match:

a+b+c+d /: a+d -> xb + c + x

The replacing rule can contain any kind of pattern:

{a, b, 1, 2, 3.0, 0.4, "Hello", "World"} /: ?Real -> 0{a, b, 1, 2, 0, 0, "Hello", "World"}

{1, 2, Pi, 4, -5} /: Condition( x?, x>3) :> -x{1, 2, -Pi, -4, -5}

Replace a sub-string:

"Hello World!" /: "World" -> "User""Hello User!"

Exchange multiple characters:

"AB_AABB" /: {"A"->"B", "B"->"A"}"BA_BBAA"

Possible Issues

Once a rule is matching a sub-expression, no more rules will tried for this sub-expression.

{x, y, z} /: {x->y, y->z, z->0}{y, z, 0}

RuleDelayed:> is evaluated each time of replacement, while Rule-> is evaluated before all replacements:

{x, x, x, x, x} /: x -> Random(10){0, 0, 0, 0, 0}

{x, x, x, x, x} /: x :> Random(10){1, 10, 3, 7, 9}

Patterns that are too general replace the entire expression

f(2, 3, 4) /: x? -> x^2f(2, 3, 4)^2

f(2, 3, 4) /: x?Integer -> x^2f(4, 9, 16)

Roundfunction

Function Plot

Syntax

Round(x)

gives the integer closest to x.

Details

Round rounds .5 numbers towards the nearest even integer (bankers' rounding).
If x is an exact numerical expression, the numerical approximation of x is used to establish the result.
For a complex number x, Round is applied separately to the real and imaginary part.
Round has the attribute Listable.

Examples

Round(2.9)3

Round(2.1)2

{Round(0.5), Round(1.5)}{0, 2}

Round(-13/8)-2

Round(1.2 + 3.8*I)1+4*I

Round(Log(10))2

Round(Pi*10^50)314159265358979323846264338327950288419716939937511

Processing of Overflow and Underflow:

Round(Overflow)Overflow

Round(Underflow)0

Rule->function

Syntax

Rule(lhs, rhs)lhs -> rhs

represents a rule that transforms lhs by rhs.

Details

rhs is evaluated before the rule is applied.
lhs is typically a symbol or an expression with pattern objects.
Rules are typically used in Replace/: or Options.

Examples

Replace(x+y, { x->2, y->4 } )6

Replace(a+b+c+d, b+c -> 4 )4 + a + d

Replace({1, 2.0, 3, 4.0, 5}, ?Integer -> Null ){Null, 2.0, Null, 4.0, Null}

Replace({1, 2.0, 3, 4.0, 5}, x?Integer -> x^2 ){1, 2.0, 9, 4.0, 25}

Possible Issues

x = 44

Rule is evaluated immediately:

x?Integer -> x^2Pattern(x, ?Integer) -> 16

Replace({2, 3, 4}, x?Integer -> x^2 ){16, 16, 16}

RuleDelayed holds the right hand side and is evaluated after the appliement:

x?Integer :> x^2Pattern(x, ?Integer) :> x^2

Replace({2, 3, 4}, x?Integer :> x^2 ){4, 9, 16}

RuleDelayed:>function

Syntax

RuleDelayed(lhs, rhs)lhs :> rhs

represents a rule that transforms lhs by rhs and then evaluate it.

Details

rhs remains unevaluated until the rule is applied.
lhs is typically a symbol or an expression with pattern objects.
Rules are typically used in Replace/: or Options.

Examples

rhs of RuleDelayed will evaluated during each replacement.

Replace({1, x, 2, x, 3, x}, x :> Random(1000) ){1, 357, 2, 319, 3, 997}

RuleDelayed is typically used in combination with patterns:

Replace({2, 3, 4}, x?Integer :> x^3 ){8, 27, 64}

Same===function

Syntax

Same(exp₁, exp₂, … )exp₁ === exp₂ === …

gives True if all expressions exp_i are identical, and False otherwise.

Details

Same is evaluated immediately and always gives False or True.
Same verify the expressions for exact correspondence and ignores the mathematical equality, unlike Equal==.

Examples

123 === "text"False

x === yFalse

f(5) === f(5)True

The integer 1 and the real number 1.0 are not identical, but mathematically equal.

1 === 1.0 1 == 1.0False True

Overflows are identical, but the equality is unknown.

Overflow === Overflow Overflow == OverflowTrue Overflow == Overflow

Possible Issues

Same=== is evaluated immediately, while Equal== is keep unevaluated until a decision is possible.

x === 2 Replace( x === 2 , x->2 )False False

x == 2 Replace( x == 2 , x->2 )x == 2 True

Symbol names are case sensitive!

A === aFalse

Scopefunction

Syntax

Scope(sym, body)

evaluates body with the local symbol sym.

Scope({sym₁, sym₂, … }, body)

evaluates body with the local symbols sym_i.

Details

sym can be either a single symbol or an assignment like Define= or DefineDelayed:=.
Scope replaces all symbols sym_i in body by unique symbols protected from the outer scope.
Unique symbols are specified by a $-character followed by an unique index.

Examples

Scope keeps x and y unchanges in the outer scope.

x = 123; a=3; b=44

Scope({x, y}, x=a+b; y=a-b; x/y )-7

{x, y, a, b}{123, y, 3, 4}

Define temporary symbols:

Scope({z=2, w=3}, z*w )6

{z, w}{z, w}

Define temporary functions:

Scope(f(x?) := x^2, {f(5), f(y)} ){25, y^2}

f(5)f(5)

Possible Issues

Temporary symbols used in the scope will become a unique symbol if they are returned.

Scope(x, x+y )x$4 + y

Secfunction

Function Plot

Syntax

Sec(z)

gives the secant of z.

Details

Sec(z) = 1 / Cos(z).
Sec gives an exact result when possible.
If z is an exact numerical value, Sec remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Sec has the attribute Listable.

Examples

Sec(2.0)-2.403

Calculate(Sec(2), 70)-2.402997961722380989754600401420066226245121093154526013044400970212074

Sec(2.0+3.0*I)-0.041675+0.0906111*I

Particular arguments and their result:

Sec(0)1

Sec(Pi/2)ComplexInfinity

Sec(Pi/3)2

Sec(Pi/4)2^(1/2)

Pure imaginary arguments are converted into Sech:

Sec(2*I)Sech(2)

The tangent of the arcus tangent is identity:

Sec(ArcSec(z))z

Sechfunction

Function Plot

Syntax

Sech(z)

gives the hyperbolic secant of z.

Details

Sech(z) = 1 / Cosh(z).
Sech gives an exact result when possible.
If z is an exact numerical value, Sech remains unevaluated if no exact solution was found.
Sech has the attribute Listable.

Examples

Sech(2.0)0.265802

Calculate(Sech(2), 70)0.2658022288340796921208627398198889715307826544322680697146411474667245

Sech(2.0+3.0*I)-0.263513-0.0362116*I

Particular arguments and their result:

Sech(0)1

Sech(Infinity)0

Pure imaginary arguments are converted into Sec:

Sech(2*I)Sec(2)

The hyperbolic secant of the area hyperbolic secant is identity:

Sech(ArSech(z))z

Selectfunction

Syntax

Select(list, crit)

gives all elements e_i of list for which crit(e_i) is True.

Details

list can be any expression, no necessarily a List{…}.
crit should be a function receiving an element e_i from list as argument and returning True or False.
Usually, crit is a query or an anonymous function.

Examples

Select( {1,2,3,4,5,6,7} , IsEven ) Select( {1,2,3,4,5,6,7} , IsPrime ){2, 4, 6} {2, 3, 5, 7}

Select( {"Hello", x+y, 4, 3+I, "World!", z} , Matching(#, ?String) & ){"Hello", "World!"}

check(x?) := x^2 < 6Null

Select( {1, 2, 3, 4, y, 2.2}, check ){1, 2, 2.2}

Select works with any kind of expression.

Select( 1 + w - Pi + x^2 + E, IsNumeric )1 + E - Pi

Sequencefunction

Syntax

Sequence(arg₁, arg₂, … )

is equivalent to a sequence of arguments arg_i in an expression or in pattern matching.

Details

A Sequence is typically returned from AnySequence?? or AnyNullSequence??? after pattern matching.
Sequence is flatten as soon as it is a function's argument. The attribute SequenceHold can prevent this behavior.
Sequence() is equivalent to no present argument.

Examples

func(Sequence(a, b, c), d)func(a, b, c, d)

arg = Sequence(2,4,6) Plus(arg) Times(arg)Sequence(2, 4, 6) 12 48

Sequence as a returned value of a pattern matching:

f(1, 2, 3) /: f(x??) :> xSequence(1, 2, 3)

Sequence as a replacement for nothing:

f(1, 2, 3) /: 2 -> Sequence()f(1, 3)

SequenceHoldattribute

Syntax

SequenceHold

is an attribute for symbols to indicate that a Sequence is not expanded during the evaluation of the arguments.

Details

Usually, a Sequence expression is expanded during evaluation. SequenceHold can prevent such expansion.
Attributes can be set and get with Attributes.

Examples

Functions like Define= and Output have the attribute SequenceHold to prevent sequence expansion.

Attributes(Define){HoldFirst, Locked, Protected, SequenceHold}

Prevent the expansion of a Sequence before it is passed:

Attributes(h) = {SequenceHold}{SequenceHold}

h(x?, y?:0) := {{x}, {y}} /* h has the attribute SequenceHold */ f(x?, y?:0) := {{x}, {y}} /* f hasn't the attribute SequenceHold */Null

h(Sequence(1, 2)) /* Sequence(1, 2) holds and is passed as x */ f(Sequence(1, 2)) /* Sequence(1, 2) is expanded and is passed as x and y */{{1, 2}, {0}} {{1}, {2}}

SetAttributesfunction

Syntax

SetAttributes(s, attr)

adds the attribute attr for symbol s.

SetAttributes(s, {attr₁, attr₂, … } )

adds the attributes attr_i for symbol s.

Details

SetAttributes adds attributes without overwriting other attributes, unlike Attributes(s) = attr.
SetAttributes gives a list of all added attributes.
See Attributes for available attributes attr.

Examples

Add the attribute Flat to f:

f(1, f(2, 3))f(1, f(2, 3))

SetAttributes(f, Flat){Flat}

f(1, f(2, 3))f(1, 2, 3)

Add other attributes to f:

SetAttributes(f, {Orderless, HoldAll}){HoldAll, Orderless}

Now, f has three attributes:

Attributes(f){Flat, HoldAll, Orderless}

Possible Issues

Attributes can't be added to locked symbols.

SetAttributes(DirectedInfinity, Orderless)Kernel.ErrorRepetition: Attributes.Locked was called multiple times. No more reports are sent.{}

ShiftLeftfunction

Syntax

ShiftLeft(z)

shifts the bits in z one bit to the left.

ShiftLeft(z, k)

shifts the bits in z by k bits to the left.

Details and Options

z is handled on its binary form and can be numerical as well as a string.
For a complex number z, ShiftLeft is applied separately to the real and imaginary part.
Negative values of k shift to the right.
The following option can used:
Rationalize Handling of integers when bits would be dropped. The default value is False.

Examples

ShiftLeft(32)64

ShiftLeft(32, 4)512

ShiftLeft(32, -2)8

ShiftLeft(2.5, 2)10.0

ShiftLeft(2/7, 3)16/7

ShiftLeft('ABC', 4)'\x10$4'

The option Rationalize can be used to convert integers into rationals:

ShiftLeft(6, -2)1

ShiftLeft(6, -2, Rationalize->True)3/2

Possible Issues

On memory level, ShiftLeft shifts the bits to higher positions, effectively to the right:

ShiftLeft('ABCDEFGH', 32)'\0\0\0\0ABCD'

ShiftRightfunction

Syntax

ShiftRight(z)

shifts the bits in z one bit to the right.

ShiftRight(z, k)

shifts the bits in z by k bits to the right.

Details and Options

z is handled on its binary form and can be numerical as well as a string.
For a complex number z, ShiftRight is applied separately to the real and imaginary part.
Negative values of k shift to the left.
The following option can used:
Rationalize Handling of integers when bits would be dropped. The default value is False.

Examples

ShiftRight(32)16

ShiftRight(32, 4)2

ShiftRight(32, -2)128

ShiftRight(2.5, 2)0.625

ShiftRight(2/7, 3)1/28

ShiftRight('ABC', 4)'$4\x04'

The option Rationalize can be used to convert integers into rationals:

ShiftRight(6, 2)1

ShiftRight(6, 2, Rationalize->True)3/2

Possible Issues

On memory level, ShiftRight shifts the bits to lower positions, effectively to the left:

ShiftRight('ABCDEFGH', 32)'EFGH\0\0\0\0'

Signfunction

Function Plot

Syntax

Sign(z)

gives the sign of the (complex) number z.

Details

Sign is suitable for both symbolic and numerical manipulation.
In case of real numbers, Sign gives -1, 0, or +1, if z is negative, zero, or positive, respectively.
In case of complex numbers, Sign(z) is defined as z/Abs(z).
If z is an exact numerical expression, the numerical estimation of z is used to decide about the sign.
Sign has the attribute Listable.
For special cases of z, some simplifications are performed.

Examples

Sign(-123)-1

Sign(2.5)1

Sign(1.2 + 3.4*I)0.33282+0.94299*I

Sign(1+I)(1+I)/2^(1/2)

Sign uses the numerical estimation of an expression to decide about the sign of the result.

Sign(Pi-E)1

Sign(Sin(4))-1

Sin(4) // Calculate-0.756802

Simplifications are performed if possible.

Sign(-2*x)-Sign(x)

Sinfunction

Function Plot

Syntax

Sin(z)

gives the sine of z.

Details

Sin gives an exact result when possible.
If z is an exact numerical value, Sin remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Sin has the attribute Listable.

Examples

Sin(2.0)0.909297

Calculate(Sin(2), 70)0.9092974268256816953960198659117448427022549714478902683789730115309673

Sin(2.0+3.0*I)9.1545-4.16891*I

Particular arguments and their result:

Sin(0)0

Sin(Pi/2)1

Sin(Pi/3)1/2*3^(1/2)

Sin(Pi/4)1/2^(1/2)

The sine of the arcus sine is identity:

Sin(ArcSin(z))z

Sinhfunction

Function Plot

Syntax

Sinh(z)

gives the hyperbolic sine of z.

Details

Sinh gives an exact result when possible.
If z is an exact numerical value, Sinh remains unevaluated if no exact solution was found.
Sinh has the attribute Listable.

Examples

Sinh(2.0)3.62686

Calculate(Sinh(2), 70)3.626860407847018767668213982801261704886342012321135721309484474934250

Sinh(2.0+3.0*I)-3.59056+0.530921*I

Particular arguments and their result:

Sinh(0)0

The hyperbolic sine of the area hyperbolic sine is identity:

Sinh(ArSinh(z))z

Slot#function

Syntax

Slot(n)#n

represents n^th argument of an anonymous function (Function&).

#

represents the first argument of an anonymous function.

Details

Slot is only used in Function&.

Examples

Select( {0,1,2,3,4,5,6,7,8,9} , #<5 & ){0, 1, 2, 3, 4}

f = Function(#1^#2) f(a, b) f(3, 4)Function(#1^#2) a^b 81

Sortfunction

Syntax

Sort(list)

sorts the elements of list into canonical order.

Sort(list, cmp)

uses the comparison function cmp for ordering.

Details

Sort is usually used on List{…}, but handles all type of expressions.
Sort orders Integern, Rationala/b and Realx numbers by their numerical value, and Complexx+y*I numbers by their real part and then by their imaginary part.
Sort orders String"…",'…' using the Unicode collation algorithm.
Sort orders numerical values before String"…",'…', Symbols and Expressionf(…).
The optional comparison function cmp receives two arguments e₁ and e₂ (a pair of elements of list) and has to return one of the following values:
True e₁ and e₂ are in order
False e₁ and e₂ are out of order
1 e₁ comes before e₂
-1 e₁ comes after e₂
0 e₁ and e₂ are identical

Examples

Sort({ "Hello", x+y, 4, 3+I, z}){3+I, 4, "Hello", x + y, z}

Sort( f(3,1,2) )f(1, 2, 3)

Use a comparison function for custom sorting.

Sort( {-3, 2, -1, 0, Pi, 4} ){-3, -1, 0, 2, 4, Pi}

Sort( {-3, 2, -1, 0, Pi, 4}, GreaterEqual ){4, Pi, 2, 0, -1, -3}

Sort( {-3, 2, -1, 0, Pi, 4}, Abs(#1) <= Abs(#2) & ){0, -1, 2, -3, Pi, 4}

During the sort of strings the unicode collation algorithm is used.

Sort({"NINO", "Nina", "Nino", "Ninu", "Niño"}){"Nina", "Nino", "NINO", "Niño", "Ninu"}

Sorting strings by their length:

Sort({"Hello", "Foo", "Baa", "World", "!"}, Length(#1) <= Length(#2) & ){"!", "Foo", "Baa", "Hello", "World"}

Span..function

Syntax

Span(m, n)m .. n

represents a span between the integers m and n.

Span(m, n, s)m .. n .. s

represents a span between the integers m and n in steps of s.

Details

Negative integers for m or n count from the end in the respective scope.
A negative integer for s count backwards.
Span is typically used in Part[…].

Examples

"Hello World!"[7..-1]"World!"

"Hello World!"[-1..-6..-1]"!dlroW"

{a, b, c, d, e}[1..5..2]{a, c, e}

Splitfunction

Syntax

Split(str, del)

splits the string str into a list of sub-strings separated by the delimiter del.

Split(exp, el)

splits the expression exp into sub-expressions separated by the element el.

Details

If del is an empty string ("" or ''), str is splitted into single characters.
If str or del is a binary string, the result is a list of binary strings.
Split can work with any kind of expressions, not only lists.

Examples

Split( "Grumpy wizards make toxic brew for the evil Queen and Jack" , " " ){"Grumpy", "wizards", "make", "toxic", "brew", "for", "the", "evil", "Queen", "and", "Jack"}

Split( "Hello\r\nWorld!" , "\r\n" ){"Hello", "World!"}

Split( {1, 2, 0, 3, 4, 0, 5, 6} , 0 ){{1, 2}, {3, 4}, {5, 6}}

Split( h(a,x,b,c,x,d,e,f) , x )h(h(a), h(b, c), h(d, e, f))

Sqrtfunction

Function Plot

Syntax

Sqrt(z)

gives the square root of z.

Details

Sqrt(z) is converted to Power(z, 1/2).
Sqrt has the attribute Listable.

Examples

Sqrt(16)4

Sqrt(2.0)1.41421

Sqrt(-0.5)0.0+0.707107*I

Sqrt({16, 100, -4}){4, 10, 2*I}

Particular arguments and their result:

Sqrt(0)0

Sqrt(-1)I

Sqrt(Infinity)Infinity

String"…",'…'object

Syntax

"…"

is a text string, a sequence of unicode characters in double quote marks.

'…'

is a binary string, a sequence of bytes in single quote marks.

String

is the head of a string.

Details and Options

Text strings are stored in UTF-8 format.
In the sequence the \-character is used to mask the following special and unprintable characters:
\0 NUL \f FF
\a BEL \r CR
\b BS \" "
\t TAB \' '
\n LF \\ \
\v VT \x00 – \xFF Two-digit hexadecimal code
Strings can be contain the null-character.
Strings can be converted with ToString between text form and binary form or any kind of object.

Examples

Text strings with unicode characters:

"abc αβγ \t ♦♥♠♣ \0 ﬂ""abc αβγ \t ♦♥♠♣ \0 ﬂ"

Head("abc αβγ \t ♦♥♠♣ \0 ﬂ")String

Binary strings with bytes are shown as ASCII characters:

'\0\0ÿ7)<''\0\0ÿ7)<'

Head('\0\0ÿ7)<')String

Possible Issues

Strings are not added with Plus+. They must be concatenated with Join~.

"Hello " + "World!""Hello " + "World!"

"Hello " ~ "World!""Hello World!"

Text strings will be converted into binary strings, if they are joined with a binary string.

'Cost: ' ~ "10 €"'Cost: 10 â\x82¬'

SubDefine@=function

Syntax

SubDefine(lhs, rhs)lhs @= rhs

evaluates the expression rhs, assign the result as a replacement for the expression lhs, and associates it with symbols occur in the first level of lhs.

Details

lhs is typically an expression with pattern objects.
For SubDefine the assignment f(g) @= rhs is associate with g, while f(g) = rhs (Define=) is associate with f.
The expression rhs will evaluated before assignment.
If lhs and rhs are lists with same length, the assignment is done with corresponding elements.
lhs can contain Condition/? to specialize the definition.
Clear can be used to remove all assignment rules from a symbol.

Examples

An assignment with SubDefine is associate with the arguments of the expression:

f(beta) @= 22

Definitions(f)/* no definitions */Null

Definitions(beta)f(beta) @= 2Null

{ f(alpha), f(beta), f(123) }{f(alpha), 2, f(123)}

Assign a special addition for special symbols:

off + off @= off; off + on @= on; on + on @= off;Null

on + off on + off + onon off

Similar definitions are not always possible with Define= due to protected symbols.

off + on = on;Define.Protected: Plus is protected!Null

Possible Issues

An assignment with SubDefine@= has a higher priority than an assignment with Define=.

f(alpha) = 1 f(alpha) @= 21 2

f(alpha)2

SubDefineDelayed@:=function

Syntax

SubDefineDelayed(lhs, rhs)lhs @:= rhs

assigns the unevaluated expression rhs as a replacement for the expression lhs and associates it with symbols occur in the first level of lhs.

Details

lhs is typically an expression with pattern objects.
For SubDefineDelayed the assignment f(g(x)) @:= rhs is associate with g, while f(g(x)) := rhs (DefineDelayed:=) is associate with f.
The expression rhs will assigned unevaluated and is always evaluated when lhs is called.
If lhs and rhs are lists with same length, the assignment is done with corresponding elements.
lhs can contain Condition/? to specialize the definition.
Clear can be used to remove all assignment rules from a symbol.

Examples

An assignment with SubDefineDelayed is associate with the arguments of the expression:

f(a(x?)) @:= x^2Null

Definitions(f)/* no definitions */Null

Definitions(a)f(a(Pattern(x, Any))) @:= x^2Null

{ f(7), f(a(7)), f(a) }{f(7), 49, f(a)}

Assign a special addition for angles:

angle(x?) + angle(y?) @:= angle( Mod(x+y, 360) )Null

angle(270) + angle(180)angle(90)

Similar definitions are not always possible with DefineDelayed:= due to protected symbols.

angle(x?) + angle(y?) := angle( Mod(x+y, 360) )DefineDelayed.Protected: Plus is protected!Null

Possible Issues

An assignment with SubDefineDelayed@:= has a higher priority than an assignment with DefineDelayed:=.

f(g(x?)) := x^2 f(g(x?)) @:= x*2Null

f(g(10))20

Subtract-function

Syntax

Subtract(exp₁, exp₂)exp₁ - exp₂

subtracts expression exp₂ from exp₁.

Details

Subtract(exp₁, exp₂) is converted to Plus(exp₁, Times(-1, exp₂)).
Subtract is suitable for both symbolic and numerical manipulation.
Subtract has the attribute Listable.

Examples

3 - 21

2*x - 3*x-x

2 - 1/4 2 - 0.257/4 1.75

Symbol`s`object

Syntax

s

with a sequence of any kind of unicode characters is a symbol, usually called variable.

Symbol

is the head of a symbol.

Details

The name of a symbol can not start with a number and can not contain operator signs.
Any kind of object can be assigned to a symbol via Define=, DefineDelayed:=, SubDefine@=, or SubDefineDelayed@:=.
The evaluation of symbols can be controlled with attributes.
The head of an expression is usually a symbol.
Symbols can be created by a string via ToSymbol if name requirements are fulfilled.

Examples

Head(var)Symbol

foo + foo2*foo

Assign numbers to a symbol:

x = 123 y = 4123 4

{ x+y, x*y, x^y }{127, 492, 228886641}

Assign expressions to a greek symbol:

φ = a*b^2a*b^2

{ φ , φ/b }{a*b^2, a*b}

Convert a string to a symbol:

ToSymbol("world")world

Possible Issues

Symbol names are case sensitive!

Var = 123123

{Var, var}{123, var}

Built-in symbols are protected against re-definitions.

True := 123 ; 2*TrueDefineDelayed.Protected: True is protected!2*True

Symbolsfunction

Syntax

Symbols()

gives a list of all defined symbols.

Symbols(str)

gives a list of defined symbols matching string str.

Details

Symbols gives all built-in symbols and all self-defined symbols.
The search string str can contain the wildcard characters ? (one character) und * (no or multiple characters).

Examples

Symbols(){Abs, Accuracy, All, Alternatives, And, Any, AnyNullSequence, AnySequence, Apply, ArCosh, ArCoth, ArCsch, ArSech, ArSinh, ArTanh, ArcCos, ArcCot, ArcCsc, ArcSec, ArcSin, ArcTan, Arg, Attributes, Bernoulli, BinaryFormat, Binomial, Boole, Calculate, Ceil, Clear, ClearAttributes, Complement, Complex, ComplexInfinity, ComplexOverflow, ComplexUnderflow, Compound, Condition, Conjugate, Contain, Cos, Cosh, Cot, Coth, Count, Csc, Csch, Define, DefineDelayed, Definitions, Denominator, Depth, DigitBase, DirectedInfinity, Divide, E, Endianness, Equal, Erf, Erfc, EulerMascheroni, Exp, Expand, Factorial, False, Fibonacci, Flat, Flatten, Floor, Function, GoldenAngle, GoldenRatio, Greater, GreaterEqual, HarmonicNumber, Head, Hold, HoldAll, HoldFirst, HoldRest, I, If, Im, Indeterminate, Infinity, Integer, InternalForm, Intersection, IsEven, IsInfinite, IsNumeric, IsOdd, IsOverflow, IsPrime, IsUnderflow, Iterate, IterateParallel, Join, Kernel, KernelID, Kernels, LaunchKernels, Length, Less, LessEqual, List, Listable, Locked, Log, MachinePrecision, Map, MapParallel, Matching, Max, Min, Minus, Mod, Nest, None, Not, Null, Numerator, NumericConstant, NumericFunction, OneIdentity, Optional, Options, Or, Order, Orderless, Output, Overflow, Part, Partition, Pattern, Pause, Pi, Plus, Power, Precision, Print, Protected, QuitKernels, Random, Rational, Rationalize, Re, Real, Replace, Round, Rule, RuleDelayed, Same, Scope, Sec, Sech, Select, Sequence, SequenceHold, SetAttributes, ShiftLeft, ShiftRight, Sign, Sin, Sinh, Slot, Sort, Span, Split, Sqrt, String, SubDefine, SubDefineDelayed, Subtract, Symbol, Symbols, Tan, Tanh, Temporary, Times, Timing, ToInteger, ToReal, ToString, ToSymbol, True, Trunc, Underflow, Unequal, Union, Unsame, Xor}

All symbols beginning with Complex:

Symbols("Complex*"){Complex, ComplexInfinity, ComplexOverflow, ComplexUnderflow}

All symbols containing Sin:

Symbols("*Sin*"){ArSinh, ArcSin, Sin, Sinh}

All symbols with two characters:

Symbols("??"){If, Im, Or, Pi, Re}

Tanfunction

Function Plot

Syntax

Tan(z)

gives the tangent of z.

Details

Tan gives an exact result when possible.
If z is an exact numerical value, Tan remains unevaluated if no exact solution was found.
z is assumed to be in radians.
Tan has the attribute Listable.

Examples

Tan(2.0)-2.18504

Calculate(Tan(2), 70)-2.185039863261518991643306102313682543432017746227663164562955869966774

Tan(2.0+3.0*I)-0.00376403+1.00324*I

Particular arguments and their result:

Tan(0)0

Tan(Pi/2)ComplexInfinity

Tan(Pi/3)3^(1/2)

Tan(Pi/4)1

The tangent of the arcus tangent is identity:

Tan(ArcTan(z))z

Tanhfunction

Function Plot

Syntax

Tanh(z)

gives the hyperbolic tangent of z.

Details

Tanh gives an exact result when possible.
If z is an exact numerical value, Tanh remains unevaluated if no exact solution was found.
Tanh has the attribute Listable.

Examples

Tanh(1.0)0.761594

Calculate(Tanh(1), 70)0.7615941559557648881194582826047935904127685972579365515968105001219532

Tanh(2.0+0.5*I)0.979941+0.030216*I

Particular arguments and their result:

Tanh(0)0

Tanh(Infinity)1

The hyperbolic tangent of the area hyperbolic tangent is identity:

Tanh(ArTanh(z))z

Times*function

Syntax

Times(exp₁, exp₂, … )exp₁ * exp₂ * …

gives the product of all expressions exp_i.

Details

Times is suitable for both symbolic and numerical manipulation.
Times has the attributes Orderless, Flat and Listable.
Times() is defined as 1.
Times(exp) is simplified to exp.

Examples

2 * 3 * 424

x^2 * x^3x^5

2 * 1/4 2 * 0.251/2 0.5

Times is flattened and the arguments are sorted.

(a * c) * (b * 4)4*a*b*c

Times threads over List{…}:

{1, 2, 3} * 10{10, 20, 30}

{1, 2, x} * {5, 10, y}{5, 20, x*y}

Timingfunction

Syntax

Timing(exp)

measures the evaluation time of exp.

Details

Timing gives a List{…} of the measured time in seconds and the evaluated result of exp.

Examples

Timing(Pause(0.3)){0.30003, Null}

Timing(Calculate(EulerMascheroni, 70)){1.55e-5, 0.5772156649015328606065120900824024310421593359399235988057672348848677}

ToIntegerfunction

Syntax

ToInteger(exp)

converts the expression exp to an integer number, if possible.

Details and Options

exp can be any expression or object like a number, a string or a symbol.
ToInteger removes the fractional part of a rational or real number, similar to Trunc.
ToInteger clears the imaginary part of a complex number, unlike Trunc.
The following options can be used if exp is a string:
DigitBase Base of the digits to parse from a text string. The default base is 10.
BinaryFormat Format, how to parse a binary string.
Endianness Byte order in a binary string.

Examples

Convert any object or expressions to an integer:

ToInteger(5/3)1

ToInteger(-3.1415)-3

ToInteger(5+2*I)5

ToInteger(E)2

ToInteger(Pi^2)9

Convert strings to an integer:

ToInteger("12345")12345

ToInteger("FF", DigitBase->16)255

ToInteger('Oa¼\0', BinaryFormat->"Integer32")12345679

Without specifying BinaryFormat, the whole binary string is interpreted as bytes of an unsigned integer:

ToInteger('\0\0\0@êítFÐ\x9C,\x9F\f')1000000000000000000000000000000

Possible Issues

Some expressions or numbers cannot be converted in an integer:

ToInteger(Overflow)Overflow

ToInteger({1,2,3})ToInteger({1, 2, 3})

ToRealfunction

Syntax

ToReal(exp)

converts the expression exp to a real number, if possible.

Details and Options

exp can be any expression or object like a number, a string or a symbol.
The real number obtain machine number precision.
ToReal clears the imaginary part of a complex number, unlike Calculate.
The following options can be used if exp is a string:
DigitBase Base of the digits to parse from a text string. The default base is 10.
BinaryFormat Format, how to parse a binary string.

Examples

Convert any object or expressions to a real number:

ToReal(123)123.0

ToReal(3/5)0.6

ToReal(5+2*I)5.0

ToReal(E)2.71828

ToReal(Pi^2)9.8696

Convert strings to a real number:

ToReal("123.45")123.45

ToReal("FF.8", DigitBase->16)255.5

ToReal('yéöB', BinaryFormat->"float32")123.456

Possible Issues

Some expressions or numbers cannot be converted into a real number:

ToReal(ComplexOverflow)Indeterminate

ToReal({1,2,3})ToReal({1, 2, 3})

ToStringfunction

Syntax

ToString(str)

gives the text string from a binary string str or vice versa.

ToString(exp)

gives the string representation of the expression exp.

Details and Options

exp can be any expression or object like a number or a symbol.
Expressions are converted into strings via the internal format.
The following option can used:
BinaryFormat Format of the string encoding.
DigitBase Base of the digits in a text string. The default base is 10.
Endianness Byte order in a binary string.

Examples

Convert any object or expressions to a string:

ToString(123)"123"

ToString(3/2)"3/2"

ToString(-3.1415)"-3.141500000000000"

ToString(123.45678901234567890123456789e12345)"1.2345678901234567890123456789e12347"

ToString(5+2*I)"5+2*I"

ToString(E)"E"

ToString(Pi^2)"Power(Pi, 2)"

Convert a text string into a binary string and vice versa with the specified format:

ToString("yä€𝄞", BinaryFormat->"UTF16")'y\0ä\0¬ 4Ø\x1EÝ'

ToString('y\0\0\0ä\0\0\0¬ \0\0\x1EÑ\x01\0', BinaryFormat->"UTF32")"yä€𝄞"

Convert a number to a text string with the specified base:

ToString(10000, DigitBase->2)"10011100010000"

ToString(10^100, DigitBase->36)"2HQBCZU2OW52BALA8LGC3S5Y9MM5TIY0VO9TKE25466GFI6AX8GS22X7KUU8L1TDS"

ToString(1024.0625, DigitBase->16)"400.1000000000"

Convert a number to a binary string with the specified format:

ToString(123456, BinaryFormat->"Integer32")'@â\x01\0'

ToString(3.1415, BinaryFormat->"Real64")'o\x12\x83ÀÊ!\t@'

ToSymbolfunction

Syntax

ToSymbol(str)

converts the string str to a symbol, if possible.

Details and Options

str must respect the rules for symbol names, see Symbols.

Examples

ToSymbol("var")var

a = 123;Null

ToSymbol("a")123

b = "al"; c = "pha"; alpha = 456;Null

b~c"alpha"

ToSymbol(b~c)456

Possible Issues

An error occurs, when the string does not meet the rules for symbol names:

ToSymbol("3a")ToSymbol.Name: "3a" contains invalid characters for a symbol name.ToSymbol("3a")

ToSymbol("no-var")ToSymbol.Name: "no-var" contains invalid characters for a symbol name.ToSymbol("no-var")

Trueconstant

Syntax

True

represents the boolean value true.

Details

True is a possible result of comparisons and logical operations.

Examples

1 == 1True

Not(True)False

Truncfunction

Function Plot

Syntax

Trunc(x)

gives the integer part of x.

Details

If x is an exact numerical expression, the numerical approximation of x is used to establish the result.
For a complex number x, Trunc is applied separately to the real and imaginary part.
Trunc has the attribute Listable.

Examples

Trunc(-2.1)-2

Trunc(2.9)2

Trunc(-13/8)-1

Trunc(1.2 + 3.8*I)1+3*I

Trunc(Log(10))2

Trunc(Pi*10^50)314159265358979323846264338327950288419716939937510

Processing of Overflow and Underflow:

Trunc(Overflow)Overflow

Trunc(Underflow)0

Trunc(-Underflow)0

Underflowconstant

Syntax

Underflow

represents an underflow in float-pointing arithmetic.

Details

The head of Underflow is Real.
Underflow is always smaller than the lowest non-negative possible real value, but always larger than 0.

Examples

Exp(-10.0^20.0)Underflow

0.0 < UnderflowTrue

Arithmetical and mathematical functions are able to process Underflow:

3.14 + Underflow3.14

Underflow * (-2)-Underflow

3.14 / UnderflowOverflow

Log(Underflow)-Overflow

Unequal!=function

Syntax

Unequal(exp₁, exp₂, … )exp₁ != exp₂ != …

gives True if all expressions exp_i are mathematically unequal, and False if any of them are equal.

Details

Unequal verify the expressions for mathematical inequality, unlike Unsame=!=.
Inexact numbers are considered as unequal if they differ in more then their last five binary digits.

Examples

1 != 2True

1 != Pi != 3.0True

The integer 1 and the real number 1.0 are mathematically not unequal, but also not identical.

1 != 1.0 1 === 1.0False False

All argument combination pairs must be unequal.

1 != 2 != 1False

Unequal stays unevaluated if inequality is unknown.

a != ba != b

Possible Issues

Inexact numbers are considered as unequal if more the last one or two decimal digits are different. This allows an easier handling with floating point numbers.

1.0000000000000000002 != 1.0000000000000000001False

1.00000000000000000020 != 1.00000000000000000010True

Unionfunction

Syntax

Union(list)

gives a sorted version of list without duplicates.

Union(list₁, list₂, … )

gives a sorted list of all distinct elements that appear in any of list_i.

Details

If the lists list_i are considered as sets, Union gives their union list₁ ∪ list₂.
Union works with any kind of expressions as long as they have the same head.

Examples

Union( {4,3,3,1} ){1, 3, 4}

Union( {4,3,3,1} , {2,4,4,5} ){1, 2, 3, 4, 5}

Union( {4,3,3,1} , {2,3.0} , {x} ){1, 2, 3, 3.0, 4, x}

Union works with any expressions, not only lists, as long as they have the same head.

Union( a + b , a + c + d )a + b + c + d

Union( f(1, 2), f(2, 3) )f(1, 2, 3)

Possible Issues

Heads have to be the same when Union is applied on expressions.

Union( f(1, 2), g(2, 3) )Union.Heads: Heads f and g are not the same.Union(f(1, 2), g(2, 3))

Unsame=!=function

Syntax

Unsame(exp₁, exp₂, … )exp₁ =!= exp₂ =!= …

gives True if all expressions exp_i are non-identical to each other, and False otherwise.

Details

Unsame is evaluated immediately and always gives False or True.
Unsame verify the expressions for exact correspondence and ignores the mathematical unequality, unlike Unequal!=.

Examples

123 =!= "text"True

x =!= yTrue

f(5) =!= f(5)False

All argument combination pairs must be non-identical.

x =!= y =!= xFalse

The integer 1 and the real number 1.0 are not identical, but mathematically equal.

1 =!= 1.0 1 != 1.0True False

Possible Issues

Unsame=!= is evaluated immediately, while Unequal!= is keep unevaluated until a decision is possible.

x =!= 2 Replace( x =!= 2 , x->2 )True True

x != 2 Replace( x != 2 , x->2 )x != 2 False

Symbol names are case sensitive!

A =!= aTrue

Xor!!function

Syntax

Xor(bool₁, bool₂, … )bool₁ !! bool₂ !! …

gives the logical XOR (exclusive OR) bool_i.

Xor(int₁, int₂, … )int₁ !! int₂ !! …

gives the binary XOR (exclusive OR) of the integers int_i.

Xor(str₁, str₂, … )str₁ !! str₂ !! …

gives the binary XOR (exclusive OR) of the strings str_i.

Details

Xor gives True if an odd number of arguments are True, and False if an even number of arguments are True, while the rest are False.
Xor() is defined as False.
Xor(exp) is simplified to exp.
Integers are handled in their two's complement form.
Strings are converted into binary form and Xor is performed bitwise.

Examples

True !! True True !! True !! TrueFalse True

12 !! 711

'BINARY' !! 'abc''#+-ARY'

Xor is simplified depending on the number of True occurs in the arguments.

True !! a > 0a <= 0

True !! a > 0 !! Truea > 0

Arithmetic Functionsoverview

Fundamental Functions

Plus+ ― gives the sum of all arguments
Minus- ― gives the arithmetic negation of a number
Subtract- ― subtracts subsequent arguments from the first argument
Times* ― gives the product of all arguments
Divide/ ― divides the first argument by the second argument

Advanced Functions

Mod ― gives the remainder on a devision
Power^ ― gives the power of two numbers
Sqrt ― gives the square root of a number
Exp ― gives the exponential of a number
Log ― gives the logarithm of a number

Assemblingoverview

Construction

Expressionf(…) ― object with a head and sub-objects as elements
List{…} ― represents a list of elements
Iterate ― gives a list of results of an iteration

Combination

Join~ ― gives a joined expression or string
Union ― gives the union set between lists
Intersection ― gives the intersection set between lists
Complement ― gives the complement set between lists

Assignment and Pattern Matchingoverview

Assignments

Define= • DefineDelayed:= • Clear ― definition of replacement rules
SubDefine@= • SubDefineDelayed@:= ― subordinate definition of replacement rules
Rule-> • RuleDelayed:> ― definition of transformation rules

Patterns

Any? • AnySequence?? • AnyNullSequence??? ― placeholder patterns
Pattern • Options ― assignment pattern
Alternatives| • Condition/? • Optional: ― extension or restriction patterns

Symbol Properties

Symbols ― list of symbols with assignments
Attributes • Definitions ― properties of a symbol assignment
SetAttributes • ClearAttributes ― changing properties of a symbol

Assignment attributes

Protected • Locked ― restriction of assignments

Evaluation attributes

HoldAll • HoldFirst • HoldRest • SequenceHold ― evaluation of arguments
Flat • Orderless • Listable ― reconstruction of arguments
NumericFunction • NumericConstant • OneIdentity ― numerical evaluation
Temporary ― storage type

Assignmentsoverview

Direct Assignments

Clear ― removes all assignment rules
Define= ― assigns an expression to an other expression
DefineDelayed:= ― assigns an expression to an other expression with delayed evaluation
SubDefine@= ― assigns subordinated an expression to an other expression
SubDefineDelayed@:= ― assigns subordinated an expression to an other expression with delayed evaluation

Transformations

Rule-> ― defines a transformation rule between expressions
RuleDelayed:> ― defines a transformation rule between expressions with delayed evaluation

Binary Functionsoverview

Bit Shift

ShiftRight ― performs a bit shift to the right
ShiftLeft ― performs a bit shift to the left

Bit Operations

And&& ― performs a locical or binary AND operation
Not! ― performs a locical or binary NOT operation
Or|| ― performs a locical or binary OR operation
Xor!! ― performs a locical or binary XOR operation

Boolean Algebraoverview

Constants

False ― represents the boolean value false
True ― represents the boolean value true

Functions

Boole ― converts a boolean value to an integer
Not! ― performs a locical or binary NOT operation
And&& ― performs a locical or binary AND operation
Or|| ― performs a locical or binary OR operation
Xor!! ― performs a locical or binary XOR operation

Comparisonoverview

Numerical Comparison

Equal== ― compares for mathematical equality
Unequal!= ― compares for mathematical unequality
Greater> ― compares regarding strictly greater
GreaterEqual>= ― compares regarding greater or equal
Less< ― compares regarding strictly less
LessEqual<= ― compares regarding less or equal

Canonical Comparison

Same=== ― compares in terms of strictly identical
Unsame=!= ― compares in terms of strictly non-identical
Order ― gives the canonical order two objects or expressions

Topological Comparison

Matching ― performs a pattern matching

Comparison and Logicoverview

Comparison

Same=== • Order ― canonical comparison
Equal== • Unequal!= ― numerial comparison
Greater> • GreaterEqual>= • Less< • LessEqual<= ― numerical inequalities
Matching ― pattern matching

Queries

Contain ― expressional queries
IsNumeric • IsPrime • IsEven • IsOdd ― numerical queries
IsInfinite • IsUnderflow • IsOverflow ― boundary queries

Boolean Algebra

False • True ― boolean constants
Boole ― converts a boolean value into an integer
Not! • And&& • Or|| • Xor!! ― boolean operators

Complex Functionsoverview

Abs ― gives the absolute value of a number
Arg ― gives the argument of a number
Sign ― gives the sign of a number
Conjugate ― gives the complex conjugate of a number
Im ― gives the imaginary part of a complex number
Re ― gives the real part of a complex number

Core Languageoverview

Object Types

Expressionf(…) ― object with a head and sub-objects as elements
Symbols ― placeholder for any kind of objects
String"…",'…' ― sequence of unicode characters or binary bytes
Integern • Rationala/b • Realx • Complexx+y*I ― number formats

Object Properties

Head • Depth • Length ― general structural properties
Contain • Count ― composition of expression
Accuracy • Precision ― numerial properties

Object Conversion

ToSymbol • ToInteger • ToReal • ToString ― convert into other objects
DigitBase • BinaryFormat • Endianness ― formatting rules

Evaluation Constructs

Compound; • Sequence • List{…} ― sequence of expressions
If • Scope ― procedural programming
Apply@@ • Map/@ • Nest ― functional programming
Function& • Slot# ― anonymous functions
Pause • Timing ― evaluation flow

Output Forms

Output • Print • InternalForm ― output formats

Parallel Computing

IterateParallel • MapParallel ― parallel evaluation
LaunchKernels • QuitKernels • Kernels • KernelID ― evaluation kernels

Credits

Acknowledgment

CSHW89, for the project idea and help with the division algorithm.
Helle, for ASM optimization in the library for arbitrary size numbers.
Jack and Nino, for their beta testing and optimization ideas.

References

Bailey D H, Borwein P B, Plouffe S (1997), "On the Rapid Computation of Various Polylogarithmic Constants", Mathematics of Computation 66 (218): 903–913
Braun J, Romberger D, Bentz H J (2017), "Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers", Notes on Number Theory and Discrete Mathematics 23(2): 54–80
Brent R P, McMillan E M (1979), "Some New Algorithms for High-Precision Computation of Euler’s constant", Math. Comput. 34: 305–312
Gourdon X, Sebah P (2004), "The Euler constant : γ", Young 1: 2n

Evaluation Constructsoverview

Expression Sequences

Sequence ― represents a sequence of arguments
Expressionf(…) ― object with a head and sub-objects as elements
List{…} ― represents a list of elements
Compound; ― evaluates all expressions in turn

Procedural Programming

If ― evaluates a specified expression depending of the condition
Scope ― evaluation using symbols in a local scope

Functional Programming

Function& ― represents anonymous function
Slot# ― represents an argument of an anonymous function
Apply@@ ― replaces the head of an expression
Map/@ ― applies a head to each element in an expression
Nest ― gives a nested expression

Evaluation Flow

Pause ― pauses for the given time
Timing ― measures the evaluation time

Extraction and Exchangeoverview

Part[…] ― gives a part of an expression or a string
All ― specifies all elements
Span.. ― represents a span between two integers
Select ― gives all elements fitting the criteria
Flatten ― flattens a list or expression and reduces the depth
Partition ― partitions an expression or string
Split ― splits an expression or a string
Replace/: ― applies replacement rules to an expression or a string

History

0.4.4

0.4.4-0002 (Aug 28, 2022)

HIGHLIGHTED NEW FEATURE: Parallel computing on multiple processor cores
IterateParallel, MapParallel, LaunchKernels, QuitKernels, Kernels, KernelID
ADDED: Timing functions: Timing, Pause
ADDED: Reciprocal trigonometric functions: Csc, Sec, Cot, ArcCsc, ArcSec, ArcCot
ADDED: Reciprocal hyperbolic functions: Csch, Sech, Coth, ArCsch, ArSech, ArCoth
ADDED: HarmonicNumber, Erf and Erfc
ADDED: IsInfinite, IsOverflow, and IsUnderflow.
ADDED: Complexx+y*I and Rationala/b as a function.
ADDED: Function plot for numerical functions in the documentation.
UPDATE: Unicode Collation Algorithm - Version 14.0.0
CHANGE: Calculate do not affect base E as well as an integer or rational exponent in Power^ to process faster powers to base E as well as rational powers.
CHANGE: SetAttributes and ClearAttributes now give only a list of the added or only the removed attributes.
CHANGE: (internal) Largest possible magnitude of reals reduced from 2^(2^63) to 2^(2^52).
FIXES: Optimized handling of exact expressions and fixed complex number handling in: Ceil, Floor, Trunc, Round, and related functions.
FIXES: Speed optimization for arbitrary precision reals in ArcSin, ArcCos, ArcTan, Sin, Cos, Exp, Log, and related functions.
FIXES: Wrong precision tracing in Plus+, ArcTan, and related functions, in combination with exact expressions.
FIXES: Bug-fixes regarding results of: Tan
FIXES: Bug-fixes regarding memory-leaks: SetAttributes, ClearAttributes, Attributes

0.4.3

0.4.3-0002 (Jun 5, 2022)

ADDED: Binary functions: ShiftLeft, ShiftRight
ADDED: Numeric queries: Accuracy, IsNumeric, IsPrime, IsEven, IsOdd
ADDED: Rational functions: Numerator, Denominator
ADDED: Count, Contain
ADDED: Unsame=!=, Flatten, Nest, Symbols
ADDED: Bernoulli, Round
ADDED: Random for strings und second parameter for count of random numbers
ADDED: Apply@@ (operator)
ADDED: Mod for complex numbers
CHANGE: Higher precedence of Map/@ operator
CHANGE: Handling of complex numbers in ToString
CHANGE: Argument definition in Scope
FIXES: Problem under Windows 7 using the DLL
FIXES: Problem with unlimited integers in x86 version fixed
FIXES: Attention of sub-defintions for flat functions like And&&, Or||, Plus+, Times*, etc.
FIXES: String and expression handling in Replace/:
FIXES: Bug-fixes regarding a memory-leak during applying of sub-definitions
FIXES: Bug-fixes regarding stability and memory-leaks: Attributes, Expand, Im, Iterate, Mod, Random, Re, Split, SubDefine@=, SubDefineDelayed@:=, ToString
FIXES: Bug-fixes regarding outputs and results of: And&&, ArSinh, ArCosh, Ceil, Definitions, Factorial!, Floor, Im, Minus-, Mod, Or||, Power^, Precision, Random, Re, Tan, ToInteger, Trunc, Xor!!
FIXES: ArcCos, ArcSin, ArcTan, Fibonacci, Mod, Sinh regarding real number handling

0.4.2

0.4.2-0001 (Sep 18, 2021)

HIGHLIGHTED NEW FEATURE: Argument-based assignments for efficient operator overload
SubDefine@=, SubDefineDelayed@:=
ADDED: Union, Intersection, Complement
ADDED: EulerMascheroni, GoldenRatio, GoldenAngle
ADDED: Expand, Fibonacci, Binomial
ADDED: Tan, Tanh, ArcCos, ArcSin, ArCosh, ArSinh, ArTanh
ADDED: Conjugate, Re, Im, Min, Max
CHANGE: Set, SetDelayed, and Table renamed to Define=, DefineDelayed:=, and Iterate, respectively
CHANGE: Print gives the last argument, instead of Null
FIXES: Completely new pattern matching engine

0.4.0

0.4.0-0005 (Mar 20, 2021)

ADDED: Support for Linux OS
FIXES: Minor bug-fixes

0.4.0-0001 (Mar 13, 2021)

HIGHLIGHTED NEW FEATURE: Support of real numbers with arbitrary precision
ADDED: Factorial!, Precision, Cos, Sin, Cosh, Sinh, ArcTan
CHANGE: Calculate - additional parameter for precision

0.3.8

0.3.8-0012 (Nov 21, 2020)

ADDED: Support of x86 processors
ADDED: Two more examples for pure basic implementation
ADDED: Floor, Ceil, Trunc, Order
FIXES: Minor bug-fixes

0.3.8-0002 (Nov 15, 2020)

First publication

Hyperbolic Functionsoverview

Hyperbolic Functions

Sinh ― gives the hyperbolic sine of a number
Cosh ― gives the hyperbolic cosine of a number
Tanh ― gives the hyperbolic tangent of a number

Reciprocal Hyperbolic Functions

Csch ― gives the hyperbolic cosecant of a number
Sech ― gives the hyperbolic secant of a number
Coth ― gives the hyperbolic cotangent of a number

Inverse Hyperbolic Functions

ArSinh ― gives the area hyperbolic sine of a number
ArCosh ― gives the area hyperbolic cosine of a number
ArTanh ― gives the area hyperbolic tangent of a number

Inverse Reciprocal Hyperbolic Functions

ArCsch ― gives the area hyperbolic cosecant of a number
ArSech ― gives the area hyperbolic secant of a number
ArCoth ― gives the area hyperbolic cotangent of a number

Introduction

Getting started

The syntax of Lizard is organized in six different types of input tokens:

Numbers	`123` , `3.14`	a sequence of decimal digits with an optional decimal point
Operators	`+` , `!=`, `->`	individual defined character groups
Symbols	`x` , `φ`, `var`	a sequence of any unicode characters which are not defined as operator
Strings	`"yä€𝄞"` , `'\0\0\0ÿ'`	a sequence of unicode or binary characters
Expressions	`head(arg1, arg2)`	a function with a head and arguments
Comments	`/* no code */`	not interpreted code blocks

In Lizard all operators are just the short form of their functional definition.
This means, you can either enter calculations in operator form like 2 + 3 ^ 5 or in functional form like Plus(2, Power(3, 5)).
Lizard is a symbolic computation scripting language.
This means, you can either perform a numerical calculation like 1 + 2 * 3 which will give 7,
or a symbolic evaluation like 2*a - 3*a which will give -a.
Everything which can not evaluated into a simple object type will be kept and handled as a general symbolic expression.
Most of the built-in functions of Lizard working also with such general symbolic expressions.
Lizard uses polymorphic functions.
This means, functions like Random() give different results depending on parameter type and count.
Many functions in Lizard use optional arguments or named parameters in form like option -> value.
This allows a specification of functions without a restricted sequence of arguments.
Lizard keeps calculation exact as long as exact inputs are given.
This means, an input like Cos(Pi/6) gives 1/2*3^(1/2), while an input like Cos(Pi/6.0) gives 0.866025.
To perform fast numerical calculations it is advised to use floating point numbers.
Lizard performs calculations first of all with fast processor word sized (64-bit) integers as well as floating point numbers.
Only if larger integers or floating point numbers with huge magnitudes or higher precisions are needed, slower arbitrary size and arbitrary precision arithmetic is performed.

Highlighted features

Full support of symbolic expression.
Support of arbitrary size integers.
Support of real numbers with arbitrary precision.
Support of rational and complex numbers.
Full unicode character string and binary string support.
Support of parallel computing using multiple cores.
Support of polymorphic or anonymous functions.
Powerful pattern matching engine for assignments and replacements.
Overloading of built-in functions for custom definitions.
Support of named parameters in functions.
High flexibility in work with subparts of strings, lists or general expressions.
Numerous conversion functions, supporting different number formats and string formats under consideration of endianness and digit base.

Lists and Expressionsoverview

Assembling

Expressionf(…) • List{…} • Iterate ― constucting lists or expressions
Join~ • Union • Intersection • Complement ― combining lists or expressions

Extraction and Exchange

Part[…] • Span.. • All • Select ― extracting elements
Partition • Split ― subdividing elements
Replace/: ― exchange elements

Mathematicsoverview

Objects, Constants and Boundaries

Numbers Types

Integern ― exact integer number
Rationala/b ― fractional number with numerator and denominator
Realx ― inexact real number with arbitrary precision
Complexx+y*I ― complex number with real and imaginary part

Numerical Constants

I ― represents the imaginary unit
Pi • E • EulerMascheroni ― fundamental constants
GoldenRatio • GoldenAngle ― other constants

Numerical Boundaries

Underflow • Overflow ― boundaries of real numbers
ComplexUnderflow • ComplexOverflow ― boundaries of complex numbers
Infinity • ComplexInfinity • DirectedInfinity ― mathematical limits
Indeterminate ― numerical indeterminate

Mathematical Functions

Arithmetic Functions

Plus+ • Subtract- • Minus- ― first level of arithmetic
Times* • Divide/ • Mod ― second level of arithmetic
Power^ • Exp • Log • Sqrt ― third level of arithmetic

Numerical Functions

Calculate ― numerial evaluation of an expression
Abs • Sign ― properties of real numbers
Ceil • Floor • Trunc • Round ― convert real numbers into integer numbers
Min • Max ― order of numbers

Complex Functions

I ― imaginary unit
Abs • Arg ― properties of complex numbers
Conjugate • Re • Im ― convert complex numbers

Trigonometric Functions

Sin • Cos • Tan • Csc • Sec • Cot ― trigonometric functions
ArcSin • ArcCos • ArcTan • ArcCsc • ArcSec • ArcCot ― inverse trigonometric functions

Hyperbolic Functions

Sinh • Cosh • Tanh • Csch • Sech • Coth ― hyperbolic functions
ArSinh • ArCosh • ArTanh • ArCsch • ArSech • ArCoth ― inverse hyperbolic functions

Binary Functions

ShiftLeft • ShiftRight ― bit shift
And&& • Not! • Or|| • Xor!! ― bit operations

Special Functions

Factorial! • Binomial ― combinatoric functions
Fibonacci • Bernoulli • HarmonicNumber ― number series
Erf • Erfc • non-elementary integral functions
Random ― random numbers
Expand ― expands out products and integer powers

Numerical Boundariesoverview

Indeterminate ― represents a mathematical quantity whose magnitude is undetermined

Real Boundaries

Underflow ― represents an numerical underflow
Overflow ― represents an numerical overflow

Complex Boundaries

ComplexUnderflow ― represents an underflow in complex numbers
ComplexOverflow ― represents an overflow in complex numbers

Infinities

Infinity ― mathematical positive infinite quantity
ComplexInfinity ― mathematical infinite quantity with an undetermined complex phase
DirectedInfinity ― mathematical infinite quantity with an known complex phase

Numerical Constantsoverview

I ― represents the imaginary unit
E ― represents the constant e ≅ 2.71828
Pi ― represents the constant π ≅ 3.14159
EulerMascheroni ― represents the constant γ ≅ 0.577216
GoldenRatio ― represents the golden ratio φ ≅ 1.61803
GoldenAngle ― represents the golden angle ≅ 137.5°

Numerical Functionsoverview

Calculate ― gives an approximate numerical value
Sign ― gives the sign of a number
Abs ― gives the absolute value of a number
Min ― gives the smallest number
Max ― gives the largest number

Rounding Functions

Ceil ― rounds toward positive infinity
Floor ― rounds toward negative infinity
Trunc ― rounds toward zero
Round ― rounds toward the nearest integer

Bit Functions

ShiftRight ― performs a bit shift to the right
ShiftLeft ― performs a bit shift to the left

Rational Functions

Numerator ― gives the numerator of an expression
Denominator ― gives the denominator of an expression

Object Conversionoverview

Functions

ToInteger ― converts an expression to an integer number
ToReal ― converts an expression to a real number
ToString ― gives the string representation of an expression
ToSymbol ― converts a string to a symbol

Option Rules

DigitBase ― is an option to define the digit base for number representations
BinaryFormat ― is an option to define the binary format for numeric and string representations
Endianness ― is an option to define the endianness

Object Propertiesoverview

Head ― gives the head of an expression
Depth ― gives nesting depth of an expression
Length ― gives the length of an expression or a string
Contain ― query for containing elements
Count ― count for matching elements or strings

Numerical properties

Precision ― gives the number of precise decimal digits of the number
Accuracy ― gives the number of precise decimal digits right of the decimal point

Object Typesoverview

Expressionf(…) ― object with a head and sub-objects as elements
Symbols ― placeholder for any kind of objects
String"…",'…' ― sequence of unicode characters or binary bytes

Numerical types

Integern ― exact integer number
Rationala/b ― fractional number with numerator and denominator
Realx ― inexact real number with arbitrary precision
Complexx+y*I ― complex number with real and imaginary part

Output Formsoverview

Output ― evaluates all expressions in turn and pass them to the output stream
Print ― evaluates all expressions in turn and pass them to the print stream
InternalForm ― gives the internal representation of an expression without operator signs

Parallel Computingoverview

Evaluation Kernels

KernelID ― give the unique ID of the processing kernel
Kernels ― gives a list of IDs of all parallel launched kernels.
LaunchKernels ― launches kernels for parallel computing.
QuitKernels ― quits parallel launched kernels.

Parallel Evaluation

IterateParallel ― gives a list of results of a parallel iteration
MapParallel ― applies a head in parallel to each element in an expression

Patternsoverview

Placeholders

Any? ― matches to any object
AnySequence?? ― matches to a sequence of any objects
AnyNullSequence??? ― matches to a sequence of any or no objects

Assignments

Pattern ― assigns a matching object to a symbol
Options ― matches a sequence of rules for symbols

Supplements

Alternatives| ― matches to all specified patterns
Condition/? ― only matches if the condition mets
Optional: ― is a pattern object and represents an optional argument

Queriesoverview

Contain ― query for containing elements

Numerical Queries

IsNumeric ― query for numeric quantity
IsUnderflow ― query for an numerical underflow
IsOverflow ― query for an numerical overflow
IsInfinite ― query for an infinite expression
IsPrime ― query for a prime number
IsEven ― query for an even number
IsOdd ― query for an odd number

Special Functionsoverview

Combinatoric functions

Factorial! ― gives the factorial of a number
Binomial ― gives the binomial coefficient of two numbers

Number series

Bernoulli ― gives a Bernoulli number
Fibonacci ― gives a Fibonacci number
HarmonicNumber ― gives a harmonic number
Prime

Number theory

Zeta
Random ― gives a pseudo-random number, string or element from a list

Non-elementary integral functions

Erf ― gives the error function value of a number
Erfc ― gives the complementary error function value of a number

Arithmetic

Expand ― expands out products and positive integer powers

Stringsoverview

Construction

String"…",'…' ― sequence of unicode characters or binary bytes
Join~ ― gives a joined expression or string

Properties

Length ― gives the length of an expression or a string
Count ― count for matching elements or strings

Extraction and Exchange

Part[…] ― gives a part of an expression or a string
Replace/: ― applies replacement rules to an expression or a string
Split ― splits an expression or a string

Symbol Propertiesoverview

Symbols ― gives a list of all defined symbols
Definitions ― prints all assignments defined to a symbol
Attributes ― gives or sets attributes for a symbol
SetAttributes ― adds attributes for a symbol
ClearAttributes ― removes attributes for a symbol

Assignment attributes

Protected ― indicate that a symbol is protected against changes by assignments or clearing
Locked ― indicate that a symbol is locked against changes in its attributes

Evaluation attributes

Flat ― indicate that arguments can be flatted or sub expressions can be created during pattern matching
Orderless ― indicate that arguments can be sorted and rearranged in pattern matching
Listable ― indicate that a symbol threads into lists occurring in arguments.
HoldAll ― indicate that all symbol's arguments maintain unevaluated
HoldFirst ― indicate that the first symbol's argument maintain unevaluated
HoldRest ― indicate that the all, except the first, symbol's argument maintain unevaluated
SequenceHold ― indicate that a sequence is not expanded during evaluation of arguments

Syntax

Brackets

Expressions

`f(x)`	`f(x)`	Expression`f`(…)
`f(x)(y)`	`(f(x))(y)`	Expression`f`(…)

Part

`f[x]`	`Part(f, x)`
`f[x][y]`	`Part(Part(f, x), y)`
`f[x,y,…]`	`Part(f, x, y, …)`

Lists

{x, y, …} List(x, y, …)

Strings

`"x"`		String"…",'…'
`'x'`		String"…",'…'

Precedence of Operators and Separators

Level 21 (high)

`?h`	`Any(h)`
`??h`	`AnySequence(h)`
`???h`	`AnyNullSequence(h)`
`s?h`	`Pattern(s, Any(h))`
`s??h`	`Pattern(s, AnySequence(h))`
`s???h`	`Pattern(s, AnyNullSequence(h))`

Level 20

f @ g @ x f(g(x)) Expressionf(…)

Level 19

`f @@ g @@ x`	`Apply(f, Apply(g, x))`
`f /@ g /@ x`	`Map(f, Map(g, x))`

Level 18

`x!`	`Factorial(x)`

Level 17

x ~ y ~ … Join(x, y, …)

Level 16

x ^ y ^ z Power(x, Power(y, z))

Level 15

`x * y * …`	`Times(x, y, …)`
`x / y / z`	`Divide(Divide(x, y), z)`

Level 14

`-x`	`Minus(x)`
`x + y + …`	`Plus(x, y, …)`
`x - y - z`	`Subtract(Subtract(x, y), z)`

Level 13

`x == y == …`	`Equal(x, y, …)`
`x != y != …`	`Unequal(x, y, …)`
`x < y < …`	`Less(x, y, …)`
`x <= y <= …`	`LessEqual(x, y, …)`
`x > y > …`	`Greater(x, y, …)`
`x >= y >= …`	`GreaterEqual(x, y, …)`

Level 12

`x === y === …`	`Same(x, y, …)`
`x .. y .. …`	`Span(x, y, …)`

Level 11

!b Not(b)

Level 10

`x \|\| y \|\| …`	`Or(x, y, …)`
`x && y && …`	`And(x, y, …)`
`x !! y !! …`	`Xor(x, y, …)`

Level 9

x | y | … Alternatives(x, y, …)

Level 8

x /? y /? f Condition(x, Condition(y, f))

Level 7

x : y : z Optional(Optional(x, y), z)

Level 6

`x -> y -> z`	`Rule(x, Rule(y, z))`
`x :> y :> z`	`RuleDelayed(x, RuleDelayed(y, z))`

Level 5

x /: y /: z Replace(Replace(x, y), z))

Level 4

x // f // g f(g(x)) Expressionf(…)

Level 3

f & Function(f)

Level 2

`x = y = z`	`Define(x, Define(y, z))`
`x := y := z`	`DefineDelayed(x, DefineDelayed(y, z))`
`x @= y @= z`	`SubDefine(x, SubDefine(y, z))`
`x @:= y @:= z`	`SubDefineDelayed(x, SubDefineDelayed(y, z))`

Level 1

x ; y ; … Compound(x, y, …)

Level 0 (low)

x , y , … Sequence(x, y, …)

Trigonometric Functionsoverview

Trigonometric Functions

Sin ― gives the sine of a number
Cos ― gives the cosine of a number
Tan ― gives the tangent of a number

Reciprocal Trigonometric Functions

Csc ― gives the cosecant of a number
Sec ― gives the secant of a number
Cot ― gives the cotangent of a number

Inverse Trigonometric Functions

ArcSin ― gives the arcus sine of a number
ArcCos ― gives the arcus cosine of a number
ArcTan ― gives the arcus tangent of a number

Inverse Reciprocal Trigonometric Functions

ArcCsc ― gives the arcus cosecant of a number
ArcSec ― gives the arcus secant of a number
ArcCot ― gives the arcus cotangent of a number

Lizard – DocumentationKernel Version 0.4.4-0002

Core Language

Object Types ― Expressionf(…) • String"…",'…' • Integern • …
Object Properties ― Head • Depth • Length • …
Object Conversion ― ToString • ToInteger • BinaryFormat • …
Evaluation Constructs ― Compound; • If • List{…} • …
Output Forms ― Print • InternalForm • …
Parallel Computing ― LaunchKernels • IterateParallel • MapParallel • …

Numerical Constants ― Pi • E • EulerMascheroni • …
Numerical Boundaries ― Overflow • ComplexUnderflow • Infinity • …
Arithmetic Functions ― Plus+ • Times* • Power^ • …
Numerical Functions ― Abs • Floor • Max • …
Complex Functions ― Arg • Conjugate • Re • …
Trigonometric Functions ― Sin • Cot • ArcCos • …
Hyperbolic Functions ― Cosh • Sech • ArTanh • …
Binary Functions ― And&& • ShiftLeft • …
Special Functions ― Factorial! • Fibonacci • Erf • Random • …

`True`	`e₁` and `e₂` are in order
`False`	`e₁` and `e₂` are out of order
`1`	`e₁` comes before `e₂`
`-1`	`e₁` comes after `e₂`
`0`	`e₁` and `e₂` are identical

`\0`	NUL	`\f`	FF
`\a`	BEL	`\r`	CR
`\b`	BS	`\"`	"
`\t`	TAB	`\'`	'
`\n`	LF	`\\`	\
`\v`	VT	`\x00` – `\xFF`	Two-digit hexadecimal code

DigitBase	Base of the digits to parse from a text string. The default base is `10`.
BinaryFormat	Format, how to parse a binary string.
Endianness	Byte order in a binary string.

BinaryFormat	Format of the string encoding.
DigitBase	Base of the digits in a text string. The default base is `10`.
Endianness	Byte order in a binary string.

Index

Absfunction

Function Plot

Syntax

Details

Examples

See also

Accuracyfunction

Syntax

Details

Examples

See also

Allconstant

Syntax

Details

Examples

See also

Alternatives|function

Syntax

Details

Examples

See also

And&&function

Syntax

Details

Examples

See also

Any?function

Syntax

Details

Examples

Possible Issues

See also

AnyNullSequence???function

Syntax

Details

Examples

See also

AnySequence??function

Syntax

Details

Examples

See also

Apply@@function

Syntax

Details

Examples

See also

ArcCosfunction

Function Plot

Syntax

Details

Examples

See also

ArcCotfunction

Function Plot

Syntax

Details

Examples

See also

ArcCscfunction

Function Plot

Syntax

Details

Examples

See also

ArCoshfunction

Function Plot

Syntax

Details

Examples

See also

ArCothfunction

Function Plot

Syntax

Details

Examples

See also

ArCschfunction

Function Plot