(directly go to documentation on : Gamma, Zeta, Bernoulli, Euler, LambertW. )

8. Special functions

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

Gamma Euler's Gamma function
Zeta Riemann's Zeta function
Bernoulli Bernoulli numbers and polynomials
Euler Euler numbers and polynomials
LambertW Lambert's W function


Gamma -- Euler's Gamma function

Standard library
Calling format:
Gamma(x)

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. 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> Gamma(1.3)
Out> Gamma(1.3);
In> N(Gamma(1.3),30)
Out> 0.897470696306277188493754954771;
In> Gamma(1.5)
Out> Sqrt(Pi)/2;
In> N(Gamma(1.5),30);
Out> 0.88622692545275801364908374167;

See also:
! , N , gamma .


Zeta -- Riemann's Zeta function

Standard library
Calling format:
Zeta(x)

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. 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(Zeta(1.3))
Out> 3.93194921180954422697490751058798;
In> Zeta(2)
Out> Pi^2/6;
In> N(Zeta(2));
Out> 1.64493406684822643647241516664602;

See also:
! , N .


Bernoulli -- Bernoulli numbers and polynomials

Standard library
Calling format:
Bernoulli(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.

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

See also:
Bin .


LambertW -- Lambert's W function

Standard library
Calling format:
LambertW(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).

To compute the numeric value of the principal branch of Lambert's W function for real arguments x>= -Exp(-1) to current precision, one can call N(LambertW(x)) (where the function N tries to approximate its argument with a real value).

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

See also:
Exp .