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Statements
∫
0
∞
e
-
z
t
t
ν
-
1
M
κ
,
μ
(
t
)
d
t
=
Γ
(
μ
+
ν
+
1
2
)
(
z
+
1
2
)
μ
+
ν
+
1
2
F
1
2
(
1
2
+
μ
-
κ
,
1
2
+
μ
+
ν
1
+
2
μ
;
1
z
+
1
2
)
,
superscript
subscript
0
superscript
𝑒
𝑧
𝑡
superscript
𝑡
𝜈
1
Whittaker-confluent-hypergeometric-M
𝜅
𝜇
𝑡
𝑡
Euler-Gamma
𝜇
𝜈
1
2
superscript
𝑧
1
2
𝜇
𝜈
1
2
Gauss-hypergeometric-F-as-2F1
1
2
𝜇
𝜅
1
2
𝜇
𝜈
1
2
𝜇
1
𝑧
1
2
{\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}%
\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left%
(z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}%
+\mu-\kappa,\tfrac{1}{2}+\mu+\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),}}
ℜ
(
μ
+
ν
+
1
2
)
>
0
𝜇
𝜈
1
2
0
{\displaystyle{\displaystyle\Re(\mu+\nu+\tfrac{1}{2})>0}}
ℜ
z
>
1
2
𝑧
1
2
{\displaystyle{\displaystyle\Re z>\tfrac{1}{2}}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
F
1
2
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F-as-2F1
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};%
\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)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
z
𝑧
{\displaystyle{\displaystyle z}}
Identifiers
