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Gamma Function
gamma_BK(x) evaluates Euler's gamma function
\[ \Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t}\, dt, \]
the continuous extension of the factorial, \( \Gamma(n) = (n-1)! \) for positive integers \( n \). It is used internally by the variable-order Bessel routines and is re-exported from the bessels module for convenience.
Function shape
Gamma function
Domain and special values
| \( x \) | \( \Gamma(x) \) |
|---|---|
| \( 1 \) | \( 1 \) |
| positive integer \( n \) | \( (n-1)! \) |
| \( 0, -1, -2, \dots \) | simple poles ( \( \pm\infty \)) |
| \( +\infty \) | \( +\infty \) |
gamma_BK accepts a real argument; an integer-argument overload provides a factorial-optimized fast path for small integer orders.
Usage
real(BK) :: x(3), g(3)
x = [0.5_bk, 1.0_bk, 5.0_bk]
g = gamma_bk(x) ! elemental: [sqrt(pi), 1, 24]
Definition bessels.f90:17
Accuracy and performance
gamma_BK is faster than the gfortran intrinsic gamma (≈26 ns/eval vs ≈38 ns/eval) at matching accuracy. The relative error against an mpmath arbitrary-precision reference is shown below in ULPs of real64:
Gamma relative error vs reference
