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NAG Library Function Document

nag_complex_log_gamma (s14agc)

▸▿ 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_log_gamma (s14agc) returns the value of the logarithm of the gamma function

\ln Γ (z)

for complex

z

, .

2 Specification

#include <nag.h>

#include <nags.h>

Complex

nag_complex_log_gamma (Complex z, NagError *fail)

3 Description

nag_complex_log_gamma (s14agc) evaluates an approximation to the logarithm of the gamma function

\ln Γ (z)

defined for

Re (z) > 0

\ln Γ (z) = \ln \int_{0}^{\infty} e^{- t} t^{z - 1} d t

where

z = x + i y

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, \dots

is a singularity and a branch point.

nag_complex_log_gamma (s14agc) is based on the method proposed by Kölbig (1972) in which the value of

\ln Γ (z)

is computed in the different regions of the

z

plane by means of the formulae

\begin{array}{l} \ln Γ (z) & = & (z - \frac{1}{2}) \ln z - z + \frac{1}{2} \ln 2 π + z \sum_{k = 1}^{K} \frac{B_{2 k}}{2 k (2 k - 1)} z^{- 2 k} + R_{K} (z) & if ​ x \geq x_{0} \geq 0, \\ = & \ln Γ (z + n) - \ln \prod_{ν = 0}^{n - 1} (z + ν) & if ​ x_{0} > x \geq 0, \\ = & \ln π - \ln Γ (1 - z) - \ln (\sin π z) & if ​ x < 0, \end{array}

where

n = [x_{0}] - [x]

\{B_{2 k}\}

are Bernoulli numbers (see Abramowitz and Stegun (1972)) and

[x]

is the largest integer

\leq x

. Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo

2 π

The function uses the values

K = 10

and

x_{0} = 7

. The remainder term

R_{K} (z)

is discussed in Section 7.

To obtain the value of

\ln Γ (z)

when

z

is real and positive, nag_log_gamma (s14abc) 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: $fail$ – 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_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_TOO_CLOSE_INTEGER: On entry, $z . re$ is ‘too close’ to a non-positive integer when $z . im = 0.0$ : $z . re = 〈value〉$ , $nint (z . re) = 〈value〉$ .

7 Accuracy

The remainder term

R_{K} (z)

satisfies the following error bound:

\begin{array}{lcl} |R_{K} (z)| & \leq & \frac{|B_{2 K}|}{|(2 K - 1)|} z^{1 - 2 K} \\ \leq & \frac{|B_{2 K}|}{|(2 K - 1)|} x^{1 - 2 K} if ​ x \geq 0 . \end{array}

Thus

|R_{10} (7)| < 2.5 \times 10^{- 15}

and hence in theory the function is capable of achieving an accuracy of approximately

15

significant digits.

8 Parallelism and Performance

nag_complex_log_gamma (s14agc) is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates the logarithm of the gamma function

\ln Γ (z)

z = - 1.5 + 2.5 i

, and prints the results.

NAG Library Function Documentnag_complex_log_gamma (s14agc)