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Statements
∫
0
∞
J
ν
(
a
t
)
J
ν
(
b
t
)
t
d
t
t
2
-
z
2
=
{
1
2
π
i
J
ν
(
b
z
)
H
ν
(
1
)
(
a
z
)
,
a
>
b
1
2
π
i
J
ν
(
a
z
)
H
ν
(
1
)
(
b
z
)
,
b
>
a
}
,
superscript
subscript
0
Bessel-J
𝜈
𝑎
𝑡
Bessel-J
𝜈
𝑏
𝑡
𝑡
𝑡
superscript
𝑡
2
superscript
𝑧
2
1
2
𝜋
𝑖
Bessel-J
𝜈
𝑏
𝑧
Hankel-H-1-Bessel-third-kind
𝜈
𝑎
𝑧
𝑎
𝑏
1
2
𝜋
𝑖
Bessel-J
𝜈
𝑎
𝑧
Hankel-H-1-Bessel-third-kind
𝜈
𝑏
𝑧
𝑏
𝑎
{\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}%
\left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}=\left\{\begin{array}[]{ll}%
\frac{1}{2}\pi iJ_{\nu}\left(bz\right){H^{(1)}_{\nu}}\left(az\right),&a>b\\
\frac{1}{2}\pi iJ_{\nu}\left(az\right){H^{(1)}_{\nu}}\left(bz\right),&b>a\end{%
array}\right\},}}
ℜ
ν
>
-
1
,
ℑ
z
>
0
formulae-sequence
𝜈
1
𝑧
0
{\displaystyle{\displaystyle\Re\nu>-1,\Im z>0}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
H
ν
(
1
)
(
z
)
Hankel-H-1-Bessel-third-kind
𝜈
𝑧
{\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
ℑ
absent
{\displaystyle{\displaystyle\Im}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
