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Statements
H
ν
(
1
)
(
ν
+
a
ν
1
3
)
H
ν
(
2
)
(
ν
+
a
ν
1
3
)
}
∼
2
4
3
ν
1
3
e
∓
π
i
/
3
Ai
(
e
∓
π
i
/
3
2
1
3
a
)
∑
k
=
0
∞
P
k
(
a
)
ν
2
k
/
3
+
2
5
3
ν
e
±
π
i
/
3
Ai
′
(
e
∓
π
i
/
3
2
1
3
a
)
∑
k
=
0
∞
Q
k
(
a
)
ν
2
k
/
3
,
asymptotic-expansion
selection
Hankel-H-1-Bessel-third-kind
𝜈
𝜈
𝑎
superscript
𝜈
1
3
Hankel-H-2-Bessel-third-kind
𝜈
𝜈
𝑎
superscript
𝜈
1
3
superscript
2
4
3
superscript
𝜈
1
3
superscript
𝑒
minus-or-plus
𝜋
𝑖
3
Airy-Ai
superscript
𝑒
minus-or-plus
𝜋
𝑖
3
superscript
2
1
3
𝑎
superscript
subscript
𝑘
0
shorthand-Legendre-P-first-kind
𝑘
𝑎
superscript
𝜈
2
𝑘
3
superscript
2
5
3
𝜈
superscript
𝑒
plus-or-minus
𝜋
𝑖
3
diffop
Airy-Ai
1
superscript
𝑒
minus-or-plus
𝜋
𝑖
3
superscript
2
1
3
𝑎
superscript
subscript
𝑘
0
shorthand-Legendre-Q-second-kind
𝑘
𝑎
superscript
𝜈
2
𝑘
3
{\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1%
}{3}}\right)\\
{H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{%
\nu^{\frac{1}{3}}}e^{\mp\pi i/3}\mathrm{Ai}\left(e^{\mp\pi i/3}2^{\frac{1}{3}}%
a\right)\sum_{k=0}^{\infty}\frac{P_{k}\left(a\right)}{\nu^{2k/3}}+\frac{2^{%
\frac{5}{3}}}{\nu}e^{\pm\pi i/3}\mathrm{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}%
}a\right)\sum_{k=0}^{\infty}\frac{Q_{k}\left(a\right)}{\nu^{2k/3}},}}
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}}}
P
ν
(
z
)
=
P
ν
0
(
z
)
shorthand-Legendre-P-first-kind
𝜈
𝑧
Legendre-P-first-kind
0
𝜈
𝑧
{\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}%
\left(z\right)}}
Q
ν
(
z
)
=
Q
ν
0
(
z
)
shorthand-Legendre-Q-second-kind
𝜈
𝑧
Legendre-Q-second-kind
0
𝜈
𝑧
{\displaystyle{\displaystyle Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}%
\left(z\right)}}
k
𝑘
{\displaystyle{\displaystyle k}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
