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9 ϑ Functions
9.2
Jacobi ϑ Functions
9.2.1 General Equations and Computational Aspects
The ϑ functions can be defined by the following Fourier series expansion: [2]
ϑ 1 (z, τ ) = 2 q
1/4
∞
n=0
(−1)
n q
n(n+1) sin ((2n + 1)z) ,
(9.1a)
ϑ 2 (z, τ ) = 2 q
1/4
∞
n=0
q
n(n+1) cos ((2n + 1)z) ,
(9.1b)
ϑ 3 (z, τ ) = 1 + 2
∞
n=1
q
n 2 cos(2n z), and
(9.1c)
ϑ 4 (z, τ ) = 1 + 2
∞
n=0
(−1)
n q
n 2 cos(2n z)
(9.1d)
with the nome q defined by
q = exp(iπτ ), with (τ ) > 0,
(9.2)
thus 0 < |q| < 1. An example is plotted in Fig. 9.1. (Note, in some publications,
instead of z → π · z is used.)
-2
-1
0
1
2
-1
-0.5
0
0.5
1
-2
-1
0
1
2
-1
0
1
-2
-1
0
1
2
-1
0
1
-2
-1
0
1
2
-0.5
0
0.5
Fig. 9.1 This graphic visualizes the ϑ 1 (solid line), ϑ 2 (dashed line), ϑ 3 (dotted line), and ϑ 4
function (dash–dot line). Top left-hand side τ = i · 0.75, top right-hand side τ = i · 0.25, and
bottom τ = 0.25 + i · 0.25. On left-hand side, the real part of the function values, and on the righthand side, its imaginary part. (Horizontal the z-values and vertical the corresponding function
values.)