120
8 Hypergeometric Functions
Due to the “info” the computation was based on a series expansion combined
with transformation (8.5). “trafo” is the abbreviation for the used linear transformation and the number refers to the type of linear transformation: trafo3:
Eqs. (8.5) or (8.6); trafo5: Eq. (8.7); trafo6: Eq. (8.8); trafo7: Eq. (8.9); trafo8:
Eq. (8.10).
Figure (8.1) was computed with the following code:
x = linspace(-10,10,1000);
a = 0.75; b = 1.25;
c = 2.5;
%
figure
%;
subplot(2,2,1)
[obj,result] = gausshyp(a,b,c,x);
plot(x,real(obj.value),x,imag(obj.value))
ylabel(’2F1’), grid on
%
subplot(2,2,2)
[obj,result] = gausshyp(2 * a,b,c,x);
plot(x,real(obj.value),x,imag(obj.value))
grid on
%
subplot(2,2,3)
[obj,result] = gausshyp(a,b,2 * c,x);
plot(x,real(obj.value),x,imag(obj.value))
xlabel(’x’), ylabel(’2F1’), grid on
%
subplot(2,2,4)
[obj,result] = gausshyp(2 * a,b,2 * c,x);
plot(x,real(obj.value),x,imag(obj.value))
xlabel(’x’), grid on
and Fig. 8.2 via
% for c negative integer and a negative integer
%
poles occur for c > a
% parameter
cr = [-6:0.2:-2.2, -2.1:0.025:-1.9, -1.8:0.1:-1.2, ...
-1.1:0.025:-0.9, -0.9:0.1:-0.2, ...
-0.1:0.025:0.1, 0.2:0.2:2];
ci = linspace(-1.5,1.5,31);
a = -3;
b = 1.25;
z = 0.5;
%
[Cr, Ci] = meshgrid(cr,ci);
c = Cr + i * Ci;
% function evaluation
tic
for n = 1:numel(c)
erg(n) = gausshyp(a,b,c(n),z).value;
end