# NAG CL Interfaces14agc (gamma_​log_​complex)

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## 1Purpose

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

## 2Specification

 #include
 Complex s14agc (Complex z, NagError *fail)
The function may be called by the names: s14agc, nag_specfun_gamma_log_complex or nag_complex_log_gamma.

## 3Description

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
 $ln⁡Γ(z)=ln⁡∫0∞e-ttz-1dt$
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.
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
 $ln⁡Γ(z) = (z-12)ln⁡z-z+12ln⁡2π+z∑k=1K B2k2k(2k-1) z-2k+RK(z) if ​x≥x0≥0, = ln⁡Γ(z+n)-ln⁡∏ν=0 n-1(z+ν) if ​x0>x≥0, = ln⁡π-ln⁡Γ(1-z)-ln(sin⁡πz) if ​x<0,$
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, s14abc can be used.
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

## 5Arguments

1: $\mathbf{z}$Complex Input
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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_TOO_CLOSE_INTEGER
On entry, ${\mathbf{z}}\mathbf{.}\mathbf{re}$ is ‘too close’ to a non-positive integer when ${\mathbf{z}}\mathbf{.}\mathbf{im}=0.0$: ${\mathbf{z}}\mathbf{.}\mathbf{re}=⟨\mathit{\text{value}}⟩$, $\mathrm{nint}\left({\mathbf{z}}\mathbf{.}\mathbf{re}\right)=⟨\mathit{\text{value}}⟩$.

## 7Accuracy

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

## 8Parallelism and Performance

s14agc is not threaded in any implementation.

None.

## 10Example

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.

### 10.1Program Text

Program Text (s14agce.c)

### 10.2Program Data

Program Data (s14agce.d)

### 10.3Program Results

Program Results (s14agce.r)