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Statements
∫
0
∞
e
-
t
t
b
-
1
-
1
2
ν
𝐌
(
a
,
b
,
t
)
J
ν
(
2
x
t
)
d
t
=
x
-
a
+
1
2
ν
e
-
x
𝐌
(
ν
-
b
+
1
,
ν
-
a
+
1
,
x
)
,
superscript
subscript
0
superscript
𝑒
𝑡
superscript
𝑡
𝑏
1
1
2
𝜈
Kummer-confluent-hypergeometric-bold-M
𝑎
𝑏
𝑡
Bessel-J
𝜈
2
𝑥
𝑡
𝑡
superscript
𝑥
𝑎
1
2
𝜈
superscript
𝑒
𝑥
Kummer-confluent-hypergeometric-bold-M
𝜈
𝑏
1
𝜈
𝑎
1
𝑥
{\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{%
\mathbf{M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=x^{-a+%
\frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(\nu-b+1,\nu-a+1,x\right),}}
2
ℜ
a
<
ℜ
ν
+
5
2
2
𝑎
𝜈
5
2
{\displaystyle{\displaystyle 2\Re a<\Re\nu+\tfrac{5}{2}}}
ℜ
b
>
0
𝑏
0
{\displaystyle{\displaystyle\Re b>0}}
x
>
0
𝑥
0
{\displaystyle{\displaystyle x>0}}
2
ℜ
a
<
ℜ
ν
+
5
2
2
𝑎
𝜈
5
2
{\displaystyle{\displaystyle 2\Re a<\Re\nu+\tfrac{5}{2}}}
ℜ
b
>
0
𝑏
0
{\displaystyle{\displaystyle\Re b>0}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\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}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
