The function may be called by the names: s14agc, nag_specfun_gamma_log_complex or nag_complex_log_gamma.
s14agc evaluates an approximation to the logarithm of the gamma function defined for by
where is complex. It is extended to the rest of the complex plane by analytic continuation unless , in which case is real and each of the points is a singularity and a branch point.
s14agc is based on the method proposed by Kölbig (1972) in which the value of is computed in the different regions of the plane by means of the formulae
where , are Bernoulli numbers (see Abramowitz and Stegun (1972)) and is the largest integer . Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo .
The function uses the values and . The remainder term is discussed in Section 7.
To obtain the value of when is real and positive, s14abc can be used.
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
1: – ComplexInput
On entry: the argument of the function.
must not be ‘too close’ (see Section 6) to a non-positive integer when .
2: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
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.
On entry, is ‘too close’ to a non-positive integer when : , .
The remainder term satisfies the following error bound:
Thus and hence in theory the function is capable of achieving an accuracy of approximately significant digits.
8Parallelism and Performance
s14agc is not threaded in any implementation.
This example evaluates the logarithm of the gamma function at , and prints the results.