naginterfaces.library.specfun.gamma_​log_​scaled_​real

naginterfaces.library.specfun.gamma_log_scaled_real(x)

gamma_log_scaled_real returns the value of $ln\left(G\left(x\right)\right)$, the scaled logarithm of the gamma function $\Gamma \left(x\right)$.

For full information please refer to the NAG Library document for s14ah

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s14ahf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

slgxfloat

The value of $ln\left(G\left(x\right)\right)$, the scaled logarithm of the gamma function $\Gamma \left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x\le 0.0$. On failure, the function value returned is zero.

Notes

gamma_log_scaled_real calculates an approximate value for $ln\left(G\left(x\right)\right)$, where $G\left(x\right)=\Gamma (x+1)/{\left(\frac{x}{e}\right)}^x$. This is a variant of the $ln\left(\Gamma \left(x\right)\right)$ function (see also gamma_log_real()), which avoids rounding problems for very large arguments by computing $ln\left(\Gamma \left(x\right)\right)$ with the Stirling approximation factored out.

For $0<x<15$, $ln\left(G\left(x\right)\right)=ln\left(\Gamma (x+1)\right)-xln\left(x\right)+x$;

and for $15\le x$, $ln\left(G\left(x\right)\right)=\frac{1}{2}ln\left(x\right)+ln\left(\sqrt{2\pi }\right)+\frac{1}{x}R\left(1/x^2\right)$, where $R$ is a suitable Remez approximation.

For $x\le 0.0$, the value $ln\left(G\left(x\right)\right)$ is undefined; gamma_log_scaled_real returns zero and exits with $errno$ = 1.

References

NIST Digital Library of Mathematical Functions