tgamma, tgammaf, tgammal — true gamma function

## Synopsis

**#include <math.h>**

**double tgamma(double** *x***);****float tgammaf(float** *x***);****long double tgammal(long double** *x***);**

`Link with `

*-lm*.

Feature Test Macro Requirements for glibc (see feature_test_macros(7)):

**tgamma**(), **tgammaf**(), **tgammal**():

_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L

## 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.

## See Also

## Colophon

This page is part of release 5.04 of the Linux *man-pages* project. A description of the project, information about reporting bugs, and the latest version of this page, can be found at https://www.kernel.org/doc/man-pages/.

## Referenced By

The man pages tgammaf(3) and tgammal(3) are aliases of tgamma(3).