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Statements
U
(
a
,
b
,
x
)
=
2
e
1
2
x
Γ
(
a
)
(
2
β
tanh
(
w
2
)
(
1
-
e
-
w
β
)
-
b
β
1
-
b
K
1
-
b
(
2
β
a
)
+
a
-
1
(
a
-
1
+
β
1
+
β
)
1
-
b
e
-
2
β
a
O
(
1
)
)
,
Kummer-confluent-hypergeometric-U
𝑎
𝑏
𝑥
2
superscript
𝑒
1
2
𝑥
Euler-Gamma
𝑎
2
𝛽
𝑤
2
superscript
1
superscript
𝑒
𝑤
𝛽
𝑏
superscript
𝛽
1
𝑏
modified-Bessel-second-kind
1
𝑏
2
𝛽
𝑎
superscript
𝑎
1
superscript
superscript
𝑎
1
𝛽
1
𝛽
1
𝑏
superscript
𝑒
2
𝛽
𝑎
Big-O
1
{\displaystyle{\displaystyle U\left(a,b,x\right)=\frac{2e^{\frac{1}{2}x}}{%
\Gamma\left(a\right)}\left(\sqrt{\frac{2}{\beta}\tanh\left(\frac{w}{2}\right)}%
\left(\frac{1-e^{-w}}{\beta}\right)^{-b}\beta^{1-b}K_{1-b}\left(2\beta a\right%
)+a^{-1}\left(\frac{a^{-1}+\beta}{1+\beta}\right)^{1-b}e^{-2\beta a}O\left(1%
\right)\right),}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
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)}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
tanh
z
𝑧
{\displaystyle{\displaystyle\tanh\NVar{z}}}
K
ν
(
z
)
modified-Bessel-second-kind
𝜈
𝑧
{\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
