Gamma, GammaNum , Zeta, ZetaNum , Bernoulli, BernoulliArray , Euler , LambertW, LambertWNum , gamma , GoldenRatio , Catalan .

Special functions

In this chapter, special and transcendental mathematical functions are described.

Gamma, GammaNum Euler's Gamma function
Zeta, ZetaNum Riemann's Zeta function
Bernoulli, BernoulliArray Bernoulli numbers and polynomials
Euler Euler numbers and polynomials
LambertW, LambertWNum Lambert's W function
gamma Euler's constant gamma
GoldenRatio the Golden Ratio
Catalan Catalan's Constant

Gamma, GammaNum -- Euler's Gamma function

Standard library
Calling format: 
Gamma(x)
GammaNum(number)

Parameters:
x -- expression

number -- expression that can be evaluated to a number

Description:
Gamma(x) is an interface to Euler's Gamma function Gamma(x). It returns exact values on integer and half-integer arguments. GammaNum(x) or equivalently N(Gamma(x) takes a numeric parameter and always returns a floating-point number in the current precision.

Note that Euler's constant gamma<=>0.57722 is the lowercase gamma in Yacas.

Examples: 
In> Precision(30)
Out> True;
In> Gamma(1.3)
Out> Gamma(1.3);
In> N(%)
Out> 0.897470696306277188493754954771;
In> Gamma(1.5)
Out> Sqrt(Pi)/2;
In> GammaNum(1.5);
Out> 0.88622692545275801364908374167;

See also:
! , N , gamma .

Zeta, ZetaNum -- Riemann's Zeta function

Standard library
Calling format: 
Zeta(x)
ZetaNum(number)

Parameters:
x -- expression

number -- expression that can be evaluated to a number

Description:
Zeta(x) is an interface to Riemann's Zeta function zeta(s). It returns exact values on integer and half-integer arguments. ZetaNum(x) or equivalently N(Zeta(x) takes a numeric parameter and always returns a floating-point number in the current precision.

Examples: 
In> Precision(30)
Out> True;
In> Zeta(1)
Out> Infinity;
In> Zeta(1.3)
Out> Zeta(1.3);
In> N(%)
Out> 3.93194921180954422697490751058798;
In> Zeta(2)
Out> Pi^2/6;
In> ZetaNum(2);
Out> 1.64493406684822643647241516664602;

See also:
! , N .

Bernoulli, BernoulliArray -- Bernoulli numbers and polynomials

Standard library
Calling format: 
Bernoulli(index)
BernoulliArray(index)
Bernoulli(index, x)

Parameters:
x -- expression that will be the variable in the polynomial

index -- expression that can be evaluated to an integer

Description:
Bernoulli(n) evaluates the n-th Bernoulli number. Bernoulli(n, x) returns the n-th Bernoulli polynomial in the variable x. The polynomial is returned in the Horner form.

An auxiliary function BernoulliArray(n) might be useful too: it returns an array (of type GenericArray) of Bernoulli numbers up to n. The array is 1-based, so that the n-th Bernoulli number is BernoulliArray(n)[n+1].

Example: 
In> Bernoulli(20);
Out> -174611/330;
In> Bernoulli(4, x);
Out> ((x-2)*x+1)*x^2-1/30;

See also:
Gamma , Zeta .

Euler -- Euler numbers and polynomials

Standard library
Calling format: 
Euler(index)
Euler(index,x)

Parameters:
x -- expression that will be the variable in the polynomial

index -- expression that can be evaluated to an integer

Description:
Euler(n) evaluates the n-th Euler number. Euler(n,x) returns the n-th Euler polynomial in the variable x.

Examples: 
In> Euler(6)
Out> -61;
In> Euler(5,x)
Out> (x-1/2)^5+(-10*(x-1/2)^3)/4+(25*(x-1/2))/16;
In> Simplify(%)
Out> (2*x^5-5*x^4+5*x^2-1)/2;

See also:
Bin .

LambertW, LambertWNum -- Lambert's W function

Standard library
Calling format: 
LambertW(x)
LambertWNum(x)
Parameters:
x -- expression, argument of the function

Description:
Lambert's W function is (a multiple-valued, complex function) defined for any (complex) z by

W(z)*Exp(W(z))=z.

This function is sometimes useful to represent solutions of transcendental equations. For example, the equation Ln(x)=3*x can be "solved" by writing x= -3*W(-1/3). It is also possible to take a derivative or integrate this function "explicitly".

For real arguments x, W(x) is real if x>= -Exp(-1).

LambertWNum is an auxiliary function that computes the numerical value of the principal branch of Lambert's W function for real arguments x>= -Exp(-1) to current precision.

Examples: 
In> LambertW(0)
Out> 0;
In> N(LambertW(-0.24/Sqrt(3*Pi)))
Out> -0.0851224014;

See also:
Exp .

gamma -- Euler's constant gamma

Standard library
Calling format: 
N(gamma)
gamma()

Description:
These functions compute Euler's constant gamma<=>0.57722...

The constant is available symbolically as gamma or numerically as a function gamma(). This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as N(gamma). The low-level numerical computations are performed by the routine GammaConstNum.

Note that Euler's Gamma function Gamma(x) is the capitalized Gamma in Yacas.

Examples: 
In> gamma+Pi
Out> gamma+Pi;
In> N(%)
Out> 3.7188083184;
In> [Precision(20);V(gamma());]
CachedConstant: Info: constant gamma is being
  recalculated at precision 20 
GammaConstNum: Info: used 56 iterations at
  working precision 24 
Out> 0.57721566490153286061;

See also:
Gamma , N , CachedConstant .

GoldenRatio -- the Golden Ratio

Standard library
Calling format: 
N(GoldenRatio)
GoldenRatio()

Description:
These functions compute the Golden Ratio GoldenRatio<=>1.6180339887<=>(1+Sqrt(5))/2.

The constant is available symbolically as GoldenRatio or numerically as a function GoldenRatio(). This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as N(GoldenRatio).

Examples: 
In> GoldenRatio - 1
Out> GoldenRatio-1;
In> N(%)
Out> 0.6180339887;
In> N(1/GoldenRatio)
Out> 0.6180339887;
In> [Precision(20);V(GoldenRatio());]
CachedConstant: Info: constant GoldenRatio is being recalculated at precision 20 
Out> 1.6180339887498948482;

See also:
N , CachedConstant .

Catalan -- Catalan's Constant

Standard library
Calling format: 
N(Catalan)
Catalan()

Description:
These functions compute Catalan's Constant Catalan<=>0.9159655941.

The constant is available symbolically as Catalan or numerically as a function Catalan(). This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as N(Catalan). The low-level numerical computations are performed by the routine CatalanConstNum.

Examples: 
In> N(Catalan)
Out> 0.9159655941;
In> DirichletBeta(2)
Out> Catalan;
In> [Precision(20);V(Catalan());]
CachedConstant: Info: constant Catalan is being recalculated at precision 20
Out> 0.91596559417721901505;

See also:
N , CachedConstant .