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Statements
∫
0
∞
J
ν
(
a
t
)
J
ν
(
b
t
)
t
λ
d
t
=
(
a
b
)
ν
Γ
(
ν
-
1
2
λ
+
1
2
)
2
λ
(
a
2
+
b
2
)
ν
-
1
2
λ
+
1
2
Γ
(
1
2
λ
+
1
2
)
𝐅
(
2
ν
+
1
-
λ
4
,
2
ν
+
3
-
λ
4
;
ν
+
1
;
4
a
2
b
2
(
a
2
+
b
2
)
2
)
,
superscript
subscript
0
Bessel-J
𝜈
𝑎
𝑡
Bessel-J
𝜈
𝑏
𝑡
superscript
𝑡
𝜆
𝑡
superscript
𝑎
𝑏
𝜈
Euler-Gamma
𝜈
1
2
𝜆
1
2
superscript
2
𝜆
superscript
superscript
𝑎
2
superscript
𝑏
2
𝜈
1
2
𝜆
1
2
Euler-Gamma
1
2
𝜆
1
2
scaled-hypergeometric-bold-F
2
𝜈
1
𝜆
4
2
𝜈
3
𝜆
4
𝜈
1
4
superscript
𝑎
2
superscript
𝑏
2
superscript
superscript
𝑎
2
superscript
𝑏
2
2
{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{%
\nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{(ab)^{\nu}\Gamma\left(\nu-%
\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{%
2}\lambda+\frac{1}{2}}\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}%
\mathbf{F}\left(\frac{2\nu+1-\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{%
4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}\right),}}
ℜ
(
2
ν
+
1
)
>
ℜ
λ
>
-
1
2
𝜈
1
𝜆
1
{\displaystyle{\displaystyle\Re(2\nu+1)>\Re\lambda>-1}}
a
≠
b
𝑎
𝑏
{\displaystyle{\displaystyle a\neq b}}
ℜ
(
2
ν
+
1
)
>
ℜ
λ
>
-
1
2
𝜈
1
𝜆
1
{\displaystyle{\displaystyle\Re(2\nu+1)>\Re\lambda>-1}}
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)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
𝐅
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
