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s14ag:  FL   CL   CPP   AD 

NAG CL Interface
s14agc (gamma_​log_​complex)

NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ag:  FL   CL   CPP   AD 

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© The Numerical Algorithms Group Ltd. 2022
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1 Purpose

s14agc returns the value of the logarithm of the gamma function ln⁡Γ(z) for complex z, .

2 Specification

#include <nag.h>
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.

3 Description

s14agc evaluates an approximation to the logarithm of the gamma function ln⁡Γ(z) defined for Re(z)>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,… is a singularity and a branch point.
s14agc is based on the method proposed by Kölbig (1972) in which the value of ln⁡Γ(z) 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=[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 RK(z) is discussed in Section 7.
To obtain the value of ln⁡Γ(z) when z is real and positive, 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 – Complex Input
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 7 in the Introduction to the NAG Library CL Interface).

6 Error 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, z.re is ‘too close’ to a non-positive integer when z.im=0.0: z.re=⟨value⟩, nint(z.re)=⟨value⟩.

7 Accuracy

The remainder term RK(z) satisfies the following error bound:
|RK(z)| ≤ |B2K| |(2K-1)| z1-2K ≤ |B2K| |(2K-1)| x1-2Kif ​x≥0.  
Thus |R10(7)|<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

s14agc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates the logarithm of the gamma function ln⁡Γ(z) 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)
NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s14ag:  FL   CL   CPP   AD 
© The Numerical Algorithms Group Ltd. 2022