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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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

Not applicable.

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)