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Statements
∫
0
∞
t
ν
+
1
J
ν
(
a
t
)
(
t
2
+
b
2
)
μ
+
1
d
t
=
a
μ
b
ν
-
μ
2
μ
Γ
(
μ
+
1
)
K
ν
-
μ
(
a
b
)
,
superscript
subscript
0
superscript
𝑡
𝜈
1
Bessel-J
𝜈
𝑎
𝑡
superscript
superscript
𝑡
2
superscript
𝑏
2
𝜇
1
𝑡
superscript
𝑎
𝜇
superscript
𝑏
𝜈
𝜇
superscript
2
𝜇
Euler-Gamma
𝜇
1
modified-Bessel-second-kind
𝜈
𝜇
𝑎
𝑏
{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at%
\right)}{(t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}%
\Gamma\left(\mu+1\right)}K_{\nu-\mu}\left(ab\right),}}
ℜ
b
>
0
𝑏
0
{\displaystyle{\displaystyle\Re b>0}}
-
1
<
ℜ
ν
<
2
ℜ
μ
+
3
2
1
𝜈
2
𝜇
3
2
{\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{2}}}
a
>
0
𝑎
0
{\displaystyle{\displaystyle a>0}}
ℜ
b
>
0
𝑏
0
{\displaystyle{\displaystyle\Re b>0}}
-
1
<
ℜ
ν
<
2
ℜ
μ
+
3
2
1
𝜈
2
𝜇
3
2
{\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{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)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
K
ν
(
z
)
modified-Bessel-second-kind
𝜈
𝑧
{\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
