Gamma, GammaNum , Zeta, ZetaNum , Bernoulli, BernoulliArray .

# 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

###

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.

##### 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 .

###

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 .