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NAG Library Function Document

nag_complex_polygamma (s14afc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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, \dots, 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) = \frac{d^{k}}{d z^{k}} ψ (z) = \frac{d^{k}}{d z^{k}} (\frac{d}{d z} \log_{e} Γ (z)),

where

z = x + i y

is complex provided

y \neq 0

and

k = 0, 1, \dots, 4

. If

y = 0

z

is real and thus

ψ^{(k)} (z)

is singular when

z = 0, - 1, - 2, \dots

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: $z$ – 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: $k$ – IntegerInput: On entry: the function $ψ^{(k)} (z)$ to be evaluated.

Constraint: $0 \leq k \leq 4$ .
3: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 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_COMPLEX: On entry, $z . re$ is ‘too close’ to a non-positive integer when $z . im = 0.0$ : $z . re = 〈value〉$ , $nint (z . re) = 〈value〉$ .
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \leq 4$ .

On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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_OVERFLOW_LIKELY: Evaluation abandoned due to likelihood of overflow.

7 Accuracy

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

8 Parallelism and Performance

nag_complex_polygamma (s14afc) is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates the psi (trigamma) function

ψ^{(1)} (z)

z = - 1.5 + 2.5 i

, and prints the results.

NAG Library Function Documentnag_complex_polygamma (s14afc)