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

Parameters

Compulsory Input Parameters

1: $z$ – complex scalar: The argument $z$ of the function.

Constraint: $Re (z)$ must not be ‘too close’ (see Error Indicators and Warnings) to a non-positive integer when $Im (z) = 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – complex scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, $Re (z)$ is ‘too close’ to a non-positive integer when $Im (z) = 0.0$ . That is, $abs (Re (z) - nint (Re (z))) < machine precision \times nint (abs (Re (z)))$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

Further Comments

None.

Example

This example evaluates the logarithm of the gamma function

\ln Γ (z)

z = - 1.5 + 2.5 i

, and prints the results.

Open in the MATLAB editor: s14ag_example

function s14ag_example


fprintf('s14ag example results\n\n');

z =  -1.5 + 2.5i;
[lgz, ifail] = s14ag(z);

disp('      z           ln(Gamma(z))');
fprintf('%5.1f%+5.1fi', real(z), imag(z));
fprintf('  %12.4e%+12.4ei\n', real(lgz), imag(lgz));

s14ag example results

      z           ln(Gamma(z))
 -1.5 +2.5i   -5.0140e+00 -4.0718e+00i

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox