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Statements
∫
0
∞
J
ν
(
b
t
)
exp
(
-
p
2
t
2
)
t
μ
-
1
d
t
=
(
1
2
b
/
p
)
ν
Γ
(
1
2
ν
+
1
2
μ
)
2
p
μ
exp
(
-
b
2
4
p
2
)
𝐌
(
1
2
ν
-
1
2
μ
+
1
,
ν
+
1
,
b
2
4
p
2
)
,
superscript
subscript
0
Bessel-J
𝜈
𝑏
𝑡
superscript
𝑝
2
superscript
𝑡
2
superscript
𝑡
𝜇
1
𝑡
superscript
1
2
𝑏
𝑝
𝜈
Euler-Gamma
1
2
𝜈
1
2
𝜇
2
superscript
𝑝
𝜇
superscript
𝑏
2
4
superscript
𝑝
2
Kummer-confluent-hypergeometric-bold-M
1
2
𝜈
1
2
𝜇
1
𝜈
1
superscript
𝑏
2
4
superscript
𝑝
2
{\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-%
p^{2}t^{2}\right)t^{\mu-1}\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left%
(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{%
2}}\right)\*{\mathbf{M}}\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^%
{2}}{4p^{2}}\right),}}
ℜ
(
μ
+
ν
)
>
0
𝜇
𝜈
0
{\displaystyle{\displaystyle\Re(\mu+\nu)>0}}
ℜ
(
p
2
)
>
0
superscript
𝑝
2
0
{\displaystyle{\displaystyle\Re(p^{2})>0}}
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)}}
𝐌
(
a
,
b
,
z
)
Kummer-confluent-hypergeometric-bold-M
𝑎
𝑏
𝑧
{\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right%
)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
exp
z
𝑧
{\displaystyle{\displaystyle\exp\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
