Special functions

(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 .