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Statements
∫
0
∞
e
-
s
t
θ
1
(
β
π
2
ℓ
|
i
π
t
ℓ
2
)
d
t
=
∫
0
∞
e
-
s
t
θ
2
(
(
1
+
β
)
π
2
ℓ
|
i
π
t
ℓ
2
)
d
t
=
-
ℓ
s
sinh
(
β
s
)
sech
(
ℓ
s
)
,
superscript
subscript
0
superscript
𝑒
𝑠
𝑡
Jacobi-theta-tau
1
𝛽
𝜋
2
ℓ
𝑖
𝜋
𝑡
superscript
ℓ
2
𝑡
superscript
subscript
0
superscript
𝑒
𝑠
𝑡
Jacobi-theta-tau
2
1
𝛽
𝜋
2
ℓ
𝑖
𝜋
𝑡
superscript
ℓ
2
𝑡
ℓ
𝑠
𝛽
𝑠
ℓ
𝑠
{\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{%
\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{%
\infty}e^{-st}\theta_{2}\left(\frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{%
\ell^{2}}\right)\mathrm{d}t=-\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}%
\right)\operatorname{sech}\left(\ell\sqrt{s}\right),}}
θ
j
(
z
|
τ
)
Jacobi-theta-tau
𝑗
𝑧
𝜏
{\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}%
\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sech
z
𝑧
{\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
s
𝑠
{\displaystyle{\displaystyle s}}
β
𝛽
{\displaystyle{\displaystyle\beta}}
ℓ
ℓ
{\displaystyle{\displaystyle\ell}}
