6
1 Gamma Functions, Beta Functions, and Related
function gammavisu
% gamma function visualized
% due to inf at negative integer maximum
% function values restricted to 35
x = [-6.2:0.1000:5.25];%+0.05;
y = -5:0.1:5;
[X,Y] = meshgrid(x,y);
% complex plane
Z = X+i * Y;
% Gamma-function
yp = gammaC(Z);
% ’PG’ method is used
Zp = abs(yp);
% abs-method alternative
%
abs(yp,’value’)
% or yp.abs(’value’)
Zp(Zp>35)=35;
surf(X,Y,Zp),axis tight, shg
fh = gcf;
fh.Position = [60 80 961 594];
ah = gca;
ytl=ah.YTickLabel;
for k=1:length(ytl)
ytlk = [ytlk,’ i’];
end
ah.YTickLabel = ytl;
xlabel(’real’),ylabel(’imag’)
title(’|\Gamma(z)|’)
1.3
The Pochhammer Symbol
Pochhammer’s symbol [1, 3] is given by
(z) w =
Γ (z + w)
Γ (z)
.
(1.8)
For “w” positive integer this is equal to z · (z + 1) · (z + 2) · · · (z + w).
The class [obj,erg] = pochlnC(z,w) returns the logarithm of the Pochhammer symbol calling the class gammalnC and using the computation based on
the Γ Lanczos-coefficients as derived by Godfrey [2]. The two inputs “z,w” are
complex arrays either of the same size or one must be a scalar. The object “obj” has
the properties “value” (function value), “inz” (first input), and “inw” (second input).
The second output “erg” is the function values in doubles.
The class pochlnC supports the methods abs for computing the absolute value,
angle to compute the phase angle in radians, conj for complex conjugation, and
imag for returning the imaginary part and real for returning the real part of the
gammalnC object. All methods have two inputs. The first one is the object and the
second one (optional) on which class property the method should be applied on,