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Statements
1
2
π
i
∫
-
∞
(
0
+
)
e
t
z
t
-
a
U
(
a
,
b
,
y
/
t
)
d
t
=
2
z
1
2
(
2
a
-
b
-
1
)
y
1
2
(
1
-
b
)
Γ
(
a
)
Γ
(
a
-
b
+
1
)
K
b
-
1
(
2
z
y
)
,
1
2
𝜋
imaginary-unit
superscript
subscript
limit-from
0
superscript
𝑒
𝑡
𝑧
superscript
𝑡
𝑎
Kummer-confluent-hypergeometric-U
𝑎
𝑏
𝑦
𝑡
𝑡
2
superscript
𝑧
1
2
2
𝑎
𝑏
1
superscript
𝑦
1
2
1
𝑏
Euler-Gamma
𝑎
Euler-Gamma
𝑎
𝑏
1
modified-Bessel-second-kind
𝑏
1
2
𝑧
𝑦
{\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz%
}t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)%
}y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}%
\left(2\sqrt{zy}\right),}}
ℜ
z
>
0
𝑧
0
{\displaystyle{\displaystyle\Re z>0}}
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)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
K
ν
(
z
)
modified-Bessel-second-kind
𝜈
𝑧
{\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
y
𝑦
{\displaystyle{\displaystyle y}}
z
𝑧
{\displaystyle{\displaystyle z}}
