tgamma, tgammaf, tgammal - true gamma function

**#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

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)

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.

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**.

These functions first appeared in glibc in version 2.1.

For an explanation of the terms used in this section, see

**attributes**(7).

Interface |
Attribute |
Value |

tgamma (), tgammaf (), tgammal () |
Thread safety |
MT-Safe |

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

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).

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.

**gamma**(3),

**lgamma**(3)