# 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`);

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.

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`

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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