## Name

tgamma, tgammaf, tgammal — true gamma function

## Synopsis

`#include <math.h>`
 ```double tgamma(``` double x`)`;

 ```float tgammaf(``` float x`)`;

 ```long double tgammal(``` long double x`)`; Note
Feature Test Macro Requirements for glibc (see feature_test_macros(7)):
`tgamma`(), `tgammaf`(), `tgammal`():
`_ISOC99_SOURCE` || `_POSIX_C_SOURCE` >= 200112L
Note Link with `−lm`.

## DESCRIPTION

These functions calculate the Gamma function of `x`.

The Gamma function is defined by

Gamma(x) = integral from 0 to infinity of t^(x−1) e^−t dt

It is defined for every real number except for nonpositive integers. For nonnegative integral `m` one has

Gamma(m+1) = m!

and, more generally, for all `x`:

Gamma(x+1) = x * Gamma(x)

Furthermore, the following is valid for all values of `x` outside the poles:

Gamma(x) * Gamma(1 − x) = PI / sin(PI * x)

## RETURN VALUE

On success, these functions return Gamma(x).

If `x` is a NaN, a NaN is returned.

If `x` is positive infinity, positive infinity is returned.

If `x` is a negative integer, or is negative infinity, a domain error occurs, and a NaN is returned.

If the result overflows, a range error occurs, and the functions return `HUGE_VAL`, `HUGE_VALF`, or `HUGE_VALL`, respectively, with the correct mathematical sign.

If the result underflows, a range error occurs, and the functions return 0, with the correct mathematical sign.

If `x` is −0 or +0, a pole error occurs, and the functions return `HUGE_VAL`, `HUGE_VALF`, or `HUGE_VALL`, respectively, with the same sign as the 0.

## ERRORS

See math_error(7) for information on how to determine whether an error has occurred when calling these functions.

The following errors can occur:

Domain error: `x` is a negative integer, or negative infinity

`errno` is set to EDOM. An invalid floating-point exception (`FE_INVALID`) is raised (but see BUGS).

Pole error: `x` is +0 or −0

`errno` is set to ERANGE. A divide-by-zero floating-point exception (`FE_DIVBYZERO`) is raised.

Range error: result overflow

`errno` is set to ERANGE. An overflow floating-point exception (`FE_OVERFLOW`) is raised.

glibc also gives the following error which is not specified in C99 or POSIX.1-2001.

Range error: result underflow

An underflow floating-point exception (`FE_UNDERFLOW`) is raised, and `errno` is set to ERANGE.

## VERSIONS

These functions first appeared in glibc in version 2.1.

## ATTRIBUTES

For an explanation of the terms used in this section, see attributes(7).

 Interface Attribute Value `tgamma`(), `tgammaf`(), `tgammal`() Thread safety MT-Safe

## CONFORMING TO

C99, POSIX.1-2001, POSIX.1-2008.

## NOTES

This function had to be called "true gamma function" since there is already a function gamma(3) that returns something else (see gamma(3) for details).

## BUGS

Before version 2.18, the glibc implementation of these functions did not set `errno` to EDOM when `x` is negative infinity.

Before glibc 2.19, the glibc implementation of these functions did not set `errno` to ERANGE on an underflow range error. `x`

In glibc versions 2.3.3 and earlier, an argument of +0 or −0 incorrectly produced a domain error (`errno` set to EDOM and an `FE_INVALID` exception raised), rather than a pole error.

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