
1554 SPECIAL FUNCTIONS AND THEIR PROPERTIES
Here, m, n = 0, ±1, ±2, . . .
Remark 30.1. The theta functions are not elliptic functions. The very good convergence of their
series allows the computation of various elliptic integrals and elliptic functions using the relations
given above in Section 30.15.1.
30.15.2 Various Relations and Formulas. Connection with Jacobi
Elliptic Functions
◮ Linear and quadratic relations.
Linear relations (first set):
ϑ
1
v +
1
2
= ϑ
2
(v), ϑ
2
v +
1
2
= −ϑ
1
(v),
ϑ
3
v +
1
2
= ϑ
4
(v), ϑ
4
v +
1
2
= ϑ
3
(v),
ϑ
1
v +
τ
2
= ie
−iπ
v+
τ
4
ϑ
4
(v), ϑ
2
v +
τ
2
= e
−iπ
v+
τ
4
ϑ
3
(v),
ϑ
3
v +
τ
2
= e
−iπ
v+
τ
4
ϑ
2
(v), ϑ
4
v +
τ
2
= ie
−iπ
v+
τ
4
ϑ
1
(v).
Linear relations (second ...