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Statements
L
n
(
α
)
(
ν
x
)
=
(
-
1
)
n
e
1
2
ν
x
2
α
-
1
2
x
1
2
α
+
1
4
(
ζ
x
-
1
)
1
4
(
Ai
(
ν
2
3
ζ
)
ν
1
3
∑
m
=
0
M
-
1
E
m
(
ζ
)
ν
2
m
+
Ai
′
(
ν
2
3
ζ
)
ν
5
3
∑
m
=
0
M
-
1
F
m
(
ζ
)
ν
2
m
+
envAi
(
ν
2
3
ζ
)
O
(
1
ν
2
M
-
2
3
)
)
,
Laguerre-polynomial-L
𝛼
𝑛
𝜈
𝑥
superscript
1
𝑛
superscript
𝑒
1
2
𝜈
𝑥
superscript
2
𝛼
1
2
superscript
𝑥
1
2
𝛼
1
4
superscript
𝜁
𝑥
1
1
4
Airy-Ai
superscript
𝜈
2
3
𝜁
superscript
𝜈
1
3
superscript
subscript
𝑚
0
𝑀
1
subscript
𝐸
𝑚
𝜁
superscript
𝜈
2
𝑚
diffop
Airy-Ai
1
superscript
𝜈
2
3
𝜁
superscript
𝜈
5
3
superscript
subscript
𝑚
0
𝑀
1
subscript
𝐹
𝑚
𝜁
superscript
𝜈
2
𝑚
envelope-Airy-Ai
superscript
𝜈
2
3
𝜁
Big-O
1
superscript
𝜈
2
𝑀
2
3
{\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{%
e^{\frac{1}{2}\nu x}}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}%
\*\left(\frac{\zeta}{x-1}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(\nu%
^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(%
\zeta)}{\nu^{2m}}+\frac{\mathrm{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{%
\frac{5}{3}}}\sum_{m=0}^{M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\mathrm{envAi}\left%
(\nu^{\frac{2}{3}}\zeta\right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)%
\right),}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
L
n
(
α
)
(
x
)
Laguerre-polynomial-L
𝛼
𝑛
𝑥
{\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
envAi
(
x
)
envelope-Airy-Ai
𝑥
{\displaystyle{\displaystyle\mathrm{envAi}\left(\NVar{x}\right)}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
ζ
𝜁
{\displaystyle{\displaystyle\zeta}}
E
m
subscript
𝐸
𝑚
{\displaystyle{\displaystyle E_{m}}}
F
m
subscript
𝐹
𝑚
{\displaystyle{\displaystyle F_{m}}}
x
𝑥
{\displaystyle{\displaystyle x}}
