128
9 ϑ Functions
Figure 9.1 was plotted via the following code fragment (visutheta.m):
x = linspace(-2,2);
tau = i * 0.75;
subplot(2,2,1), hold on
for k=1:4
% theta_1 ... _4
obj = jacobiTheta(k, x, tau);
plot(x,obj.value), axis tight
end
...
9.3
Dedekind’s η Function
Dedekind’s η function [4] is given by
η(τ ) = q
24
∞
n= ∞
(−1)
n q
3n 2 −n
2
, with q = exp(2πiτ ).
(9.10)
(Please note the “2” in the definition of q.)
The η function transforms under the projective special linear group PSL(2, Z) as
η
aτ + b
cτ + d
= b, c, d)
√
cτ + dη(τ ),
(9.11)
with (a, b, c, d) = exp(iπR(a, b, c, d)/12) and R(a, b, c, d) given by Eq. (9.9j).
Hence, the computation is similar to the computation of the θ functions and based
on the same type of transformation.
The η function can be computed with the dedeEta class. The syntax is
[obj,result] = dedeEta(tau) with the input parameter τ (complex
scalar). The output variables are the object “obj” with properties “value,” the
function value, the input parameter “tau,” and “info” with some information with
respect to the evaluation. The output variable “result” equalt η(τ ). An example is
tau = pi + 0.1 * i;
[obj,result] = dedeEta(tau);
9.4
The Jacobi Symbol
The Jacobi symbol [5]
a
n
is defined for positive odd integer n and integer a as
a
n
=
a
p 1
m 1
a
p 2
m 2
· · ·
a
p k
m k
, where n = p
m 1
1 p
m 1 s
s
· · · p
m k
k
(9.12)