nag_complex_log_gamma (s14agc) (PDF version)
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NAG C Library Manual

# NAG Library Function Documentnag_complex_log_gamma (s14agc)

## 1  Purpose

nag_complex_log_gamma (s14agc) returns the value of the logarithm of the gamma function $\mathrm{ln}\Gamma \left(z\right)$ for complex $z$, .

## 2  Specification

 #include #include
 Complex nag_complex_log_gamma (Complex z, NagError *fail)

## 3  Description

nag_complex_log_gamma (s14agc) evaluates an approximation to the logarithm of the gamma function $\mathrm{ln}\Gamma \left(z\right)$ defined for $\mathrm{Re}\left(z\right)>0$ by
 $ln⁡Γz=ln⁡∫0∞e-ttz-1dt$
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 $z=0,-1,-2,\dots \text{}$ is a singularity and a branch point.
nag_complex_log_gamma (s14agc) is based on the method proposed by Kölbig (1972) in which the value of $\mathrm{ln}\Gamma \left(z\right)$ is computed in the different regions of the $z$ plane by means of the formulae
 $ln⁡Γz = z-12ln⁡z-z+12ln⁡2π+z∑k=1K B2k2k2k-1 z-2k+RKz if ​x≥x0≥0, = ln⁡Γz+n-ln⁡∏ν=0 n-1z+ν if ​x0>x≥0, = ln⁡π-ln⁡Γ1-z-lnsin⁡πz if ​x<0,$
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 $\text{}\le x$. 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 Section 7.
To obtain the value of $\mathrm{ln}\Gamma \left(z\right)$ when $z$ is real and positive, nag_log_gamma (s14abc) can be used.

## 4  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

## 5  Arguments

1:     zComplexInput
On entry: the argument $z$ of the function.
Constraint: ${\mathbf{z}}\mathbf{.}\mathbf{re}$ must not be ‘too close’ (see Section 6) to a non-positive integer when ${\mathbf{z}}\mathbf{.}\mathbf{im}=0.0$.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_TOO_CLOSE_INTEGER
On entry, ${\mathbf{z}}\mathbf{.}\mathbf{re}$ is ‘too close’ to a non-positive integer when ${\mathbf{z}}\mathbf{.}\mathbf{im}=0.0$: ${\mathbf{z}}\mathbf{.}\mathbf{re}=〈\mathit{\text{value}}〉$, $\mathrm{nint}\left({\mathbf{z}}\mathbf{.}\mathbf{re}\right)=〈\mathit{\text{value}}〉$.

## 7  Accuracy

The remainder term ${R}_{K}\left(z\right)$ satisfies the following error bound:
 $RKz ≤ B2K 2K-1 z1-2K ≤ B2K 2K-1 x1-2Kif ​x≥0.$
Thus $\left|{R}_{10}\left(7\right)\right|<2.5×{10}^{-15}$ and hence in theory the function is capable of achieving an accuracy of approximately $15$ significant digits.

None.

## 9  Example

This example evaluates the logarithm of the gamma function $\mathrm{ln}\Gamma \left(z\right)$ at $z=-1.5+2.5i$, and prints the results.

### 9.1  Program Text

Program Text (s14agce.c)

### 9.2  Program Data

Program Data (s14agce.d)

### 9.3  Program Results

Program Results (s14agce.r)

nag_complex_log_gamma (s14agc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual