lngammacplx#

Purpose#

Returns the natural log of the Gamma function.

Format#

f = lngammacplx(z)#

Parameters:: x_cplx (NxK matrix) – data. may be complex.
Returns:: f (NxK matrix) – The natural log of the Gamma function evaluated at x_cplx.

Examples#

// Real component
xr = { 2.5 ,
       9.1 };

// Imaginary component
xi = { 3 ,
       1 };

// Create complex matrix
x_cplx = complex(xr, xi);

// Compute gamma function
lngammacplx(x_cplx);

The results after the code:

-1.47095 +   2.82262i
10.76131 +   2.15456i

Remarks#

Note that lngammacplx(z) may yield a result with a different imaginary part than ln(gammacplx(z)). This is because lngammacplx(z) returns the value of the logarithm of gamma(z) on the corresponding branch of the complex plane, while a call to ln(z) always returns a function value with an imaginary part within \([-π,π]\). Hence the imaginary part of the result can differ by a multiple of \(2*π\). However, exp(lngammacplx(z)) = gammacplx(z). This routine uses a Lanczos series approximation for the complex ln(gamma) function.

References#

1. Lanczos, SIAM JNA 1, 1964. pp. 86-96.
1. Luke, ‘’The Special … approximations,’’ 1969 pp. 29-31.
1. Luke, ‘’Algorithms … functions,’’ 1977.
1. Spouge, SIAM JNA 31, 1994. pp. 931.
1. Press, ‘’Numerical Recipes.’’
1. Chang, ‘’Computation of special functions,’’ 1996.
P. Godfrey, ‘’A note on the computation of the convergent Lanczos complex Gamma approximation.’’
Original code by Paul Godfrey