tgamma, tgammaf, tgammal - true gamma function
These functions calculate the Gamma function of
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
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).
x is a NaN, a NaN is returned.
x is positive infinity, positive infinity is returned.
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.
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:
xis a negative integer, or negative infinity
errno is set to EDOM. An invalid floating-point exception (FE_INVALID) is raised (but see BUGS).
xis +0 or -0
errno is set to ERANGE. A divide-by-zero floating-point exception (FE_DIVBYZERO) is raised.
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.
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.
C99, POSIX.1-2001, POSIX.1-2008.
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.
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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