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Statements
f
(
τ
)
=
τ
π
sinh
(
τ
π
)
Γ
(
1
2
-
μ
+
i
τ
)
Γ
(
1
2
-
μ
-
i
τ
)
∫
1
∞
P
-
1
2
+
i
τ
μ
(
x
)
g
(
x
)
d
x
,
𝑓
𝜏
𝜏
𝜋
𝜏
𝜋
Euler-Gamma
1
2
𝜇
𝑖
𝜏
Euler-Gamma
1
2
𝜇
𝑖
𝜏
superscript
subscript
1
Legendre-P-first-kind
𝜇
1
2
𝑖
𝜏
𝑥
𝑔
𝑥
𝑥
{\displaystyle{\displaystyle f(\tau)=\frac{\tau}{\pi}\sinh\left(\tau\pi\right)%
\Gamma\left(\tfrac{1}{2}-\mu+i\tau\right)\*\Gamma\left(\tfrac{1}{2}-\mu-i\tau%
\right)\int_{1}^{\infty}P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)g(x)\mathrm{%
d}x,}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
x
𝑥
{\displaystyle{\displaystyle x}}
τ
𝜏
{\displaystyle{\displaystyle\tau}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
