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Statements
∫
0
1
2
π
J
μ
(
z
sin
2
θ
)
J
ν
(
z
cos
2
θ
)
(
sin
θ
)
2
μ
+
1
(
cos
θ
)
2
ν
+
1
d
θ
=
Γ
(
μ
+
1
2
)
Γ
(
ν
+
1
2
)
J
μ
+
ν
+
1
2
(
z
)
(
8
π
z
)
1
2
Γ
(
μ
+
ν
+
1
)
,
superscript
subscript
0
1
2
𝜋
Bessel-J
𝜇
𝑧
2
𝜃
Bessel-J
𝜈
𝑧
2
𝜃
superscript
𝜃
2
𝜇
1
superscript
𝜃
2
𝜈
1
𝜃
Euler-Gamma
𝜇
1
2
Euler-Gamma
𝜈
1
2
Bessel-J
𝜇
𝜈
1
2
𝑧
superscript
8
𝜋
𝑧
1
2
Euler-Gamma
𝜇
𝜈
1
{\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}%
\theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\mu+1}(\cos%
\theta)^{2\nu+1}\mathrm{d}\theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)%
\Gamma\left(\nu+\tfrac{1}{2}\right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8%
\pi z)^{\frac{1}{2}}\Gamma\left(\mu+\nu+1\right)},}}
ℜ
μ
>
-
1
2
,
ℜ
ν
>
-
1
2
formulae-sequence
𝜇
1
2
𝜈
1
2
{\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
