This page explains the following SciPy functions for computing the gamma function and related functions.

`gamma`

`gammaln`

`gammainc`

`gamaincinv`

`gammaincc`

`gammainccinv`

`beta`

`betaln`

`betainc`

`betaincinv`

`psi`

`polygamma`

The SciPy function names correspond easily to mathematical functions. However, as noted below, there are a couple things to be aware of.

- The functions with named ending in
`ln`

do indeed take natural logarithms, but they take absolute values first. - The functions containing
`inc`

compute*regularized*incomplete functions and not simply incomplete functions.

These points will be made clear below.

## Gamma and beta functions

The SciPy functions for gamma function Γ(z) and the beta function B(x, y) are unsurprisingly called `gamma`

and `beta`

. The function `gamma(z)`

computes

and the function `beta(x, y)`

computes

## Logarithms

Often the gamma and beta functions are not as useful in practice as their logarithms. (See explanation here.) These are implemented in SciPy as `gammaln`

and `betaln`

. Note that `gammaln`

actually returns the natural logarithm of the *absolute value* of the gamma function, i.e. `gammaln(z)`

computes log| γ(z) |. Similarly `betaln(x, y)`

computes log | B(x, y) |.

## Psi and polygamma functions

The derivative of log( Γ(z) ) is denoted ψ(z) and is implemented in the `psi`

function. The *n*th derivative of ψ(z) is implemented in `psi(n, z)`

.

## Incomplete and complementary functions

Both the gamma and the beta function have “incomplete” versions. However, SciPy’s incomplete gamma function `gammainc`

corresponds to the *regularized* gamma function. Similarly, SciPy’s incomplete beta function `betainc`

corresponds to the *regularized* beta function.

The (lower) incomplete gamma function is defined by

and the upper incomplete gamma function is defined by

These are called “incomplete” because they integrate over part of the region defining the gamma function. We will see below that the incomplete beta function follows the same pattern.

The regularized versions of the upper and lower incomplete gamma functions are

and

respectively.

The SciPy function `gammainc(a, z)`

computes P(a, z) and the function `gammaincc`

computes Q(a, z). Note the extra ‘c’ in `gammaincc`

that stands for “complement.” It may not seem necessary to provide functions for both P(a, z) and its complement Q(a, z) since Q(a, z) = 1 – P(a, z). However, the complementary functions may be necessary for numerical precision. See the explanation here regarding the error function and its complement; the same reasoning applies to incomplete gamma functions.

The incomplete beta function is defined by

and the regularized incomplete beta function is defined by

The SciPy function `betainc(a, b, x)`

computes B_{x}(a, b) and the function `betaincc`

computes I_{x}(a, b).

## Inverse functions

SciPy functions ending in `inv`

compute the inverse of the corresponding function. `gammaincinv(a, y)`

returns `x`

such that `gammainc(a, x)`

equals ` y`

. Similarly `gammainccinv`

is the inverse of `gammaincc`

. Finally, `betaincinv(a, b, y)`

returns `x`

such that `betaincinv(a, b, x)`

equals `y`

.

There is an asymmetry in the SciPy implementations of the gamma and beta functions: there is no functions `betaincc`

or `betainccinv`

. These functions are unnecessary because of symmetries in the beta function. One could define `betaincc(a, b, x)`

as `betainc(b, a, 1-x)`

and `betainccinv(a, b, x)`

as `betaincinv(b, a, 1-x)`

.

## See also

Bessel functions in SciPy

Gamma function identities

Special function relationship diagram