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NAG Library Manual

# NAG Library Routine DocumentS14AGF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S14AGF returns the value of the logarithm of the gamma function $\mathrm{ln}\Gamma \left(z\right)$ for complex $z$, via the function name.

## 2  Specification

 FUNCTION S14AGF ( Z, IFAIL)
 COMPLEX (KIND=nag_wp) S14AGF
 INTEGER IFAIL COMPLEX (KIND=nag_wp) Z

## 3  Description

S14AGF 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.
S14AGF 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 routine 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, S14ABF 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  Parameters

1:     Z – COMPLEX (KIND=nag_wp)Input
On entry: the argument $z$ of the function.
Constraint: $\mathrm{Re}\left({\mathbf{Z}}\right)$ must not be ‘too close’ (see Section 6) to a non-positive integer when $\mathrm{Im}\left({\mathbf{Z}}\right)=0.0$.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, $\mathrm{Re}\left({\mathbf{Z}}\right)$ is ‘too close’ to a non-positive integer when $\mathrm{Im}\left({\mathbf{Z}}\right)=0.0$. That is, .

## 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 routine 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 (s14agfe.f90)

### 9.2  Program Data

Program Data (s14agfe.d)

### 9.3  Program Results

Program Results (s14agfe.r)