naginterfaces.library.specfun.gamma_​log_​complex

naginterfaces.library.specfun.gamma_log_complex(z)

gamma_log_complex returns the value of the logarithm of the gamma function $ln\left(\Gamma \right)\left(z\right)$ for complex $z$, .

For full information please refer to the NAG Library document for s14ag

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s14agf.html

Parameters

zcomplex

The argument $z$ of the function.

Returns

lngzcomplex

The value of $ln\left(\Gamma \right)\left(z\right)$.

Raises

NagValueError

(errno $1$)

On entry, $Re\left(z\right)$ is ‘too close’ to a non-positive integer when $Im\left(z\right)=0.0$.

Notes

gamma_log_complex evaluates an approximation to the logarithm of the gamma function $ln\left(\Gamma \right)\left(z\right)$ defined for $Re\left(z\right)>0$ by

$$ln\left(\Gamma \right)\left(z\right)=ln\left({\int }_0^{\infty }\right)e^{-t}{t}^{z-1}dt$$

where $z=x+iy$ is complex. It is extended to the rest of the complex plane by analytic continuation unless $y=0$, in which case $z$ is real and each of the points $z=0,-1,-2,\dots$ is a singularity and a branch point.

gamma_log_complex is based on the method proposed by Kölbig (1972) in which the value of $ln\left(\Gamma \right)\left(z\right)$ is computed in the different regions of the $z$ plane by means of the formulae

$$\begin{array}{c}\begin{array}{cccc}ln\left(\Gamma \right)\left(z\right)& =& (z-\frac{1}{2})ln\left(z\right)-z+\frac{1}{2}ln\left(2\right)\pi +z{\sum }_{k=1}^K\frac{B_{2k}}{2k(2k-1)}{z}^{-2k}+R_K\left(z\right)& \text{if}x\ge x_0\ge 0\text{,}\\ & =& ln\left(\Gamma \right)(z+n)-ln\left({\prod }_{\nu =0}^{n-1}\right)(z+\nu )& \text{if}{x}_0>x\ge 0\text{,}\\ & & & \\ & =& ln\left(\pi \right)-ln\left(\Gamma \right)(1-z)-ln\left(\sin \left(\pi z\right)\right)& \text{if}x<0\text{,}\end{array}\end{array}$$

where $n=\left[x_0\right]-\left[x\right]$, $\left\{B_{2k}\right\}$ are Bernoulli numbers (see Abramowitz and Stegun (1972)) and $\left[x\right]$ is the largest integer $\le x$. Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo $2\pi$.

The function uses the values $K=10$ and $x_0=7$. The remainder term $R_K\left(z\right)$ is discussed in Accuracy.

To obtain the value of $ln\left(\Gamma \right)\left(z\right)$ when $z$ is real and positive, gamma_log_real() can be used.

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Kölbig, K S, 1972, Programs for computing the logarithm of the gamma function, and the digamma function, for complex arguments, Comp. Phys. Comm. (4), 221–226