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Statements
U
(
a
,
z
)
=
e
-
1
4
z
2
z
-
a
-
1
2
2
π
i
Γ
(
1
2
+
a
)
∫
-
i
∞
i
∞
Γ
(
t
)
Γ
(
1
2
+
a
-
2
t
)
2
t
z
2
t
d
t
,
parabolic-U
𝑎
𝑧
superscript
𝑒
1
4
superscript
𝑧
2
superscript
𝑧
𝑎
1
2
2
𝜋
𝑖
Euler-Gamma
1
2
𝑎
superscript
subscript
𝑖
𝑖
Euler-Gamma
𝑡
Euler-Gamma
1
2
𝑎
2
𝑡
superscript
2
𝑡
superscript
𝑧
2
𝑡
𝑡
{\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}z^{-%
a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i%
\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}%
\mathrm{d}t,}}
|
ph
z
|
<
3
4
π
phase
𝑧
3
4
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}
a
≠
-
1
2
,
-
3
2
,
-
5
2
,
…
𝑎
1
2
3
2
5
2
…
{\displaystyle{\displaystyle a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots}}
|
ph
z
|
<
3
4
π
ph
𝑧
3
4
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\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}}
U
(
a
,
z
)
parabolic-U
𝑎
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
𝑎
{\displaystyle{\displaystyle a}}
Identifiers
