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Statements
∫
0
∞
e
-
1
2
t
t
1
2
(
ν
-
1
)
-
μ
W
κ
,
μ
(
t
)
J
ν
(
2
x
t
)
d
t
=
Γ
(
ν
-
2
μ
+
1
)
Γ
(
3
2
-
μ
-
κ
+
ν
)
e
-
1
2
x
x
1
2
(
μ
+
κ
-
3
2
)
M
1
2
(
κ
-
3
μ
+
ν
+
1
2
)
,
1
2
(
ν
-
μ
-
κ
+
1
2
)
(
x
)
,
superscript
subscript
0
superscript
𝑒
1
2
𝑡
superscript
𝑡
1
2
𝜈
1
𝜇
Whittaker-confluent-hypergeometric-W
𝜅
𝜇
𝑡
Bessel-J
𝜈
2
𝑥
𝑡
𝑡
Euler-Gamma
𝜈
2
𝜇
1
Euler-Gamma
3
2
𝜇
𝜅
𝜈
superscript
𝑒
1
2
𝑥
superscript
𝑥
1
2
𝜇
𝜅
3
2
Whittaker-confluent-hypergeometric-M
1
2
𝜅
3
𝜇
𝜈
1
2
1
2
𝜈
𝜇
𝜅
1
2
𝑥
{\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(%
\nu-1)-\mu}W_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d%
}t=\frac{\Gamma\left(\nu-2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu%
\right)}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1%
}{2}(\kappa-3\mu+\nu+\frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}%
\left(x\right),}}
max
(
2
ℜ
μ
-
1
,
-
1
)
<
ℜ
ν
2
𝜇
1
1
𝜈
{\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu}}
x
>
0
𝑥
0
{\displaystyle{\displaystyle x>0}}
max
(
2
ℜ
μ
-
1
,
-
1
)
<
ℜ
ν
2
𝜇
1
1
𝜈
{\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu}}
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
κ
,
μ
(
z
)
Whittaker-confluent-hypergeometric-M
𝜅
𝜇
𝑧
{\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
W
κ
,
μ
(
z
)
Whittaker-confluent-hypergeometric-W
𝜅
𝜇
𝑧
{\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\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
