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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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_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

Not applicable.

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)