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