nag_complex_polygamma (s14afc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_complex_polygamma (s14afc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_polygamma (s14afc) returns the value of the k th derivative of the psi function ψ z ,  for complex z  and k = 0 , 1 , , 4 .

2  Specification

#include <nag.h>
#include <nags.h>
Complex  nag_complex_polygamma (Complex z, Integer k, NagError *fail)

3  Description

nag_complex_polygamma (s14afc) evaluates an approximation to the k th derivative of the psi function ψ z  given by
ψ k z = d k dz k ψ z = d k dz k d dz log e Γ z ,
where z = x + iy  is complex provided y0  and k = 0 , 1 , , 4 . If y=0 , z  is real and thus ψ k z  is singular when z = 0 , -1 , -2 , .
Note that ψ k z  is also known as the polygamma function. Specifically, ψ 0 z  is often referred to as the digamma function and ψ 1 z  as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).
nag_complex_polygamma (s14afc) is based on a modification of the method proposed by Kölbig (1972).
To obtain the value of ψ k z  when z  is real, nag_real_polygamma (s14aec) 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: z.re  must not be ‘too close’ (see Section 6) to a non-positive integer when z.im = 0.0 .
2:     kIntegerInput
On entry: the function ψ k z  to be evaluated.
Constraint: 0 k 4 .
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_COMPLEX
On entry, z = value,value .
Constraint: z.re  must not be ‘too close’ to a non-positive integer when z.im = 0.0 . That is, z.re - nint z.re machine precision × nint z.re .
NE_INT
On entry, k=value .
Constraint: 0 k 4 .
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.
NE_OVERFLOW_LIKELY
The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.

7  Accuracy

Empirical tests have shown that the maximum relative error is a loss of approximately two decimal places of precision.

8  Further Comments

None.

9  Example

The example program evaluates the psi (trigamma) function ψ 1 z  at z = -1.5 + 2.5 i , and prints the results.

9.1  Program Text

Program Text (s14afce.c)

9.2  Program Data

Program Data (s14afce.d)

9.3  Program Results

Program Results (s14afce.r)


nag_complex_polygamma (s14afc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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