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Statements
ber
ν
′
x
∼
e
x
/
2
(
2
π
x
)
1
2
∑
k
=
0
∞
b
k
(
ν
)
x
k
cos
(
x
2
+
(
ν
2
+
3
k
4
+
1
8
)
π
)
-
1
π
(
sin
(
2
ν
π
)
ker
ν
′
x
+
cos
(
2
ν
π
)
kei
ν
′
x
)
,
asymptotic-expansion
diffop
Kelvin-ber
𝜈
1
𝑥
superscript
𝑒
𝑥
2
superscript
2
𝜋
𝑥
1
2
superscript
subscript
𝑘
0
subscript
𝑏
𝑘
𝜈
superscript
𝑥
𝑘
𝑥
2
𝜈
2
3
𝑘
4
1
8
𝜋
1
𝜋
2
𝜈
𝜋
diffop
Kelvin-ker
𝜈
1
𝑥
2
𝜈
𝜋
diffop
Kelvin-kei
𝜈
1
𝑥
{\displaystyle{\displaystyle\operatorname{ber}_{\nu}'x\sim\frac{e^{x/\sqrt{2}}%
}{(2\pi x)^{\frac{1}{2}}}\*\sum_{k=0}^{\infty}\frac{b_{k}(\nu)}{x^{k}}\cos%
\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}+\frac{1}{8}\right)%
\pi\right)-\frac{1}{\pi}(\sin\left(2\nu\pi\right)\operatorname{ker}_{\nu}'x+%
\cos\left(2\nu\pi\right)\operatorname{kei}_{\nu}'x),}}
ber
ν
(
x
)
Kelvin-ber
𝜈
𝑥
{\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
kei
ν
(
x
)
Kelvin-kei
𝜈
𝑥
{\displaystyle{\displaystyle\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
ker
ν
(
x
)
Kelvin-ker
𝜈
𝑥
{\displaystyle{\displaystyle\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
x
𝑥
{\displaystyle{\displaystyle x}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
b
k
(
ν
)
subscript
𝑏
𝑘
𝜈
{\displaystyle{\displaystyle b_{k}(\nu)}}
