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Statements
W
κ
,
μ
(
z
)
=
(
-
1
)
n
e
-
1
2
z
z
1
2
-
μ
-
n
Γ
(
1
+
2
μ
)
Γ
(
1
2
-
μ
-
κ
)
∫
0
∞
M
-
κ
,
μ
(
t
)
e
-
1
2
t
t
n
+
μ
-
1
2
t
+
z
d
t
,
Whittaker-confluent-hypergeometric-W
𝜅
𝜇
𝑧
superscript
1
𝑛
superscript
𝑒
1
2
𝑧
superscript
𝑧
1
2
𝜇
𝑛
Euler-Gamma
1
2
𝜇
Euler-Gamma
1
2
𝜇
𝜅
superscript
subscript
0
Whittaker-confluent-hypergeometric-M
𝜅
𝜇
𝑡
superscript
𝑒
1
2
𝑡
superscript
𝑡
𝑛
𝜇
1
2
𝑡
𝑧
𝑡
{\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-%
\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac%
{1}{2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right%
)e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\mathrm{d}t,}}
|
ph
z
|
<
π
phase
𝑧
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
-
ℜ
(
1
+
2
μ
)
<
n
<
|
ℜ
μ
|
+
ℜ
κ
<
1
2
1
2
𝜇
𝑛
𝜇
𝜅
1
2
{\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<%
\tfrac{1}{2}}}
|
ph
z
|
<
π
ph
𝑧
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
n
=
0
,
1
,
2
,
…
𝑛
0
1
2
…
{\displaystyle{\displaystyle n=0,1,2,\dots}}
-
ℜ
(
1
+
2
μ
)
<
n
<
|
ℜ
μ
|
+
ℜ
κ
<
1
2
1
2
𝜇
𝑛
𝜇
𝜅
1
2
{\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<%
\tfrac{1}{2}}}
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)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
∫
{\displaystyle{\displaystyle\int}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
n
𝑛
{\displaystyle{\displaystyle n}}
z
𝑧
{\displaystyle{\displaystyle z}}
