158
12 Weierstraß Functions
Examples
(a) As a first example, we compute Klein’s invariant J (τ ).
tau = pi + i;
obj = ellipWeier(’klein’, 1, tau, 0.5 + i * 0.75)
obj =
ellipWeier with properties:
value: 0.747655947903644 - 0.031588735985876i
z: NaN
para: 3.141592653589793 + 1.000000000000000i
info: 2x1 cell
obj.info
ans =
2x1 cell array
’Klein’s complete invariant J(tau)’
’parameter tau’
As an alternative, we could have used any ω 1 and ω 3 with the ratio τ =
ω 3
ω 1
omega1 = rand
omega1 =
0.814723686393179
omega3 = tau * omega1;
obj = ellipWeier(’klein’,omega1,omega3,0.5 + i * 0.75)
obj =
ellipWeier with properties:
value: 0.747655947903644 - 0.031588735985876i
z: NaN
para: 3.141592653589793 + 1.000000000000000i
info: 2x1 cell
>> obj.info
ans =
2x1 cell array
’Klein’s complete invariant J(tau)’
’parameter tau’
(b) Figure 12.1 was plotted with the following function ( visuweier.m):
% as an example we visualize some results for a
% lemniscatic lattice
g2 = 0.8;
% lattice invariant
omega1 = gamma(0.25)^2 ./ (4 * sqrt(pi) * g2^(1/4));
omega3 = i * omega1;
xl = linspace(-2.5,2.5,30);
% variable z
[X, Y] = meshgrid(xl, xl);
z = X + i * Y;
for n = 1:numel(z)
% p(z) function
res(n)=ellipWeier(’wp’,omega1,omega3,z(n)).value;
end
res = reshape(res,size(z));