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
lnΓz=ln0e-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, 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:     z ComplexInput
On entry: the argument z of the function.
Constraint: z.re must not be ‘too close’ (see Section 6) to a non-positive integer when z.im=0.0.
2:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_TOO_CLOSE_INTEGER
On entry, z.re is ‘too close’ to a non-positive integer when z.im=0.0: z.re=value, nintz.re=value.

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
Parallelism and Performance

nag_complex_log_gamma (s14agc) is not threaded in any implementation.

9
Further Comments

None.

10
Example

This example evaluates the logarithm of the gamma function lnΓz at z=-1.5+2.5i, and prints the results.

10.1
Program Text

Program Text (s14agce.c)

10.2
Program Data

Program Data (s14agce.d)

10.3
Program Results

Program Results (s14agce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017