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Statements
θ
3
′
(
z
,
q
)
θ
3
(
z
,
q
)
=
-
4
sin
(
2
z
)
∑
n
=
1
∞
q
2
n
-
1
1
+
2
q
2
n
-
1
cos
(
2
z
)
+
q
4
n
-
2
=
4
∑
n
=
1
∞
(
-
1
)
n
q
n
1
-
q
2
n
sin
(
2
n
z
)
,
diffop
Jacobi-theta
3
1
𝑧
𝑞
Jacobi-theta
3
𝑧
𝑞
4
2
𝑧
superscript
subscript
𝑛
1
superscript
𝑞
2
𝑛
1
1
2
superscript
𝑞
2
𝑛
1
2
𝑧
superscript
𝑞
4
𝑛
2
4
superscript
subscript
𝑛
1
superscript
1
𝑛
superscript
𝑞
𝑛
1
superscript
𝑞
2
𝑛
2
𝑛
𝑧
{\displaystyle{\displaystyle\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left%
(z,q\right)}=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-%
1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{%
2n}}\sin\left(2nz\right),}}
θ
j
(
z
,
q
)
Jacobi-theta
𝑗
𝑧
𝑞
{\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
n
𝑛
{\displaystyle{\displaystyle n}}
z
𝑧
{\displaystyle{\displaystyle z}}
q
𝑞
{\displaystyle{\displaystyle q}}
Identifiers
