NAG FL Interface
s14agf (gamma_​log_​complex)

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1 Purpose

s14agf returns the value of the logarithm of the gamma function lnΓ(z) for complex z, via the function name.

2 Specification

Fortran Interface
Function s14agf ( z, ifail)
Complex (Kind=nag_wp) :: s14agf
Integer, Intent (Inout) :: ifail
Complex (Kind=nag_wp), Intent (In) :: z
C Header Interface
#include <nag.h>
Complex  s14agf_ (const Complex *z, Integer *ifail)
The routine may be called by the names s14agf or nagf_specfun_gamma_log_complex.

3 Description

s14agf evaluates an approximation to the logarithm of the gamma function lnΓ(z) defined for Re(z)>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.
s14agf is based on the method proposed by Kölbig (1972) in which the value of lnΓ(z) is computed in the different regions of the z plane by means of the formulae
lnΓ(z) = (z-12)lnz-z+12ln2π+zk=1K B2k2k(2k-1) z-2k+RK(z) if ​xx00, = lnΓ(z+n)-lnν=0 n-1(z+ν) if ​x0>x0, = lnπ-lnΓ(1-z)-ln(sinπz) 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 routine uses the values K=10 and x0=7. The remainder term RK(z) is discussed in Section 7.
To obtain the value of lnΓ(z) 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 Arguments

1: z Complex (Kind=nag_wp) Input
On entry: the argument z of the function.
Constraint: Re(z) must not be ‘too close’ (see Section 6) to a non-positive integer when Im(z)=0.0.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, Re(z) is ‘too close’ to a non-positive integer when Im(z)=0.0. That is, abs(Re(z)-nint(Re(z)))<machine precision×nint(abs(Re(z))).
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The remainder term RK(z) satisfies the following error bound:
|RK(z)| |B2K| |(2K-1)| z1-2K |B2K| |(2K-1)| x1-2Kif ​x0.  
Thus |R10(7)|<2.5×10-15 and hence in theory the routine is capable of achieving an accuracy of approximately 15 significant digits.

8 Parallelism and Performance

s14agf is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s14agfe.f90)

10.2 Program Data

Program Data (s14agfe.d)

10.3 Program Results

Program Results (s14agfe.r)