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

NAG Library Function Document

nag_complex_log_gamma (s14agc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_log_gamma (s14agc) returns the value of the logarithm of the gamma function lnΓz for complex z, .

2  Specification

#include <nag.h>
#include <nags.h>
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 lnΓz defined for Rez>0 by
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, 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 lnΓz is computed in the different regions of the z plane by means of the formulae
lnΓz = z-12lnz-z+12ln2π+zk=1K B2k2k2k-1 z-2k+RKz if ​xx00, = lnΓz+n-lnν=0 n-1z+ν if ​x0>x0, = lnπ-lnΓ1-z-lnsinπz if ​x<0,
where n=x0-x, B2k are Bernoulli numbers (see Abramowitz and Stegun (1972)) and x is the largest integer x. Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo 2π.
The function uses the values K=10 and x0=7. The remainder term RKz is discussed in Section 7.
To obtain the value of lnΓz 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: must not be ‘too close’ (see Section 6) to a non-positive integer when
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
On entry, is ‘too close’ to a non-positive integer when,

7  Accuracy

The remainder term RKz satisfies the following error bound:
RKz B2K 2K-1 z1-2K B2K 2K-1 x1-2Kif ​x0.
Thus R107<2.5×10-15 and hence in theory the function is capable of achieving an accuracy of approximately 15 significant digits.

8  Further Comments


9  Example

This example evaluates the logarithm of the gamma function lnΓz 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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012