## 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`():
`_XOPEN_SOURCE` >= 600 || `_ISOC99_SOURCE` || `_POSIX_C_SOURCE` >= 200112L;
or cc `-std=c99`
Note Link with `−lm`.

## DESCRIPTION

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.

`errno` is not set for this case.

## VERSIONS

These functions first appeared in glibc in version 2.1.

## CONFORMING TO

C99, POSIX.1-2001.

## 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

If `x` is negative infinity, `errno` is not set (it should be set to EDOM).

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.

This page is part of release 3.52 of the Linux `man-pages` project. A description of the project, and information about reporting bugs, can be found at http://www.kernel.org/doc/man−pages/.