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Statements
H
ν
(
1
)
(
ν
z
)
H
ν
(
2
)
(
ν
z
)
}
∼
2
e
∓
π
i
/
3
(
4
ζ
1
-
z
2
)
1
4
(
Ai
(
e
±
2
π
i
/
3
ν
2
3
ζ
)
ν
1
3
∑
k
=
0
∞
A
k
(
ζ
)
ν
2
k
+
e
±
2
π
i
/
3
Ai
′
(
e
±
2
π
i
/
3
ν
2
3
ζ
)
ν
5
3
∑
k
=
0
∞
B
k
(
ζ
)
ν
2
k
)
,
asymptotic-expansion
selection
Hankel-H-1-Bessel-third-kind
𝜈
𝜈
𝑧
Hankel-H-2-Bessel-third-kind
𝜈
𝜈
𝑧
2
superscript
𝑒
minus-or-plus
𝜋
𝑖
3
superscript
4
𝜁
1
superscript
𝑧
2
1
4
Airy-Ai
superscript
𝑒
plus-or-minus
2
𝜋
𝑖
3
superscript
𝜈
2
3
𝜁
superscript
𝜈
1
3
superscript
subscript
𝑘
0
subscript
𝐴
𝑘
𝜁
superscript
𝜈
2
𝑘
superscript
𝑒
plus-or-minus
2
𝜋
𝑖
3
diffop
Airy-Ai
1
superscript
𝑒
plus-or-minus
2
𝜋
𝑖
3
superscript
𝜈
2
3
𝜁
superscript
𝜈
5
3
superscript
subscript
𝑘
0
subscript
𝐵
𝑘
𝜁
superscript
𝜈
2
𝑘
{\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}\left(\nu z\right)\\
{H^{(2)}_{\nu}}\left(\nu z\right)}\sim 2e^{\mp\pi i/3}\left(\frac{4\zeta}{1-z^%
{2}}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(e^{\pm 2\pi i/3}\nu^{%
\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k}(%
\zeta)}{\nu^{2k}}+\frac{e^{\pm 2\pi i/3}\mathrm{Ai}'\left(e^{\pm 2\pi i/3}\nu^%
{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B_{k}(%
\zeta)}{\nu^{2k}}\right),}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
H
ν
(
1
)
(
z
)
Hankel-H-1-Bessel-third-kind
𝜈
𝑧
{\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
H
ν
(
2
)
(
z
)
Hankel-H-2-Bessel-third-kind
𝜈
𝑧
{\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
k
𝑘
{\displaystyle{\displaystyle k}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
ζ
(
z
)
𝜁
𝑧
{\displaystyle{\displaystyle\zeta(z)}}
A
k
(
ζ
)
subscript
𝐴
𝑘
𝜁
{\displaystyle{\displaystyle A_{k}(\zeta)}}
B
k
(
ζ
)
subscript
𝐵
𝑘
𝜁
{\displaystyle{\displaystyle B_{k}(\zeta)}}
Identifiers
