Deprecated : $wgMWOAuthSharedUserIDs=false is deprecated, set $wgMWOAuthSharedUserIDs=true, $wgMWOAuthSharedUserSource='local' instead [Called from MediaWiki\HookContainer\HookContainer::run in /var/www/html/w/includes/HookContainer/HookContainer.php at line 135] in /var/www/html/w/includes/Debug/MWDebug.php on line 372 DLMF:10.20.E9 - MaRDI portal
Statements
H
ν
(
1
)
′
(
ν
z
)
H
ν
(
2
)
′
(
ν
z
)
}
∼
4
e
∓
2
π
i
/
3
z
(
1
-
z
2
4
ζ
)
1
4
(
e
∓
2
π
i
/
3
Ai
(
e
±
2
π
i
/
3
ν
2
3
ζ
)
ν
4
3
∑
k
=
0
∞
C
k
(
ζ
)
ν
2
k
+
Ai
′
(
e
±
2
π
i
/
3
ν
2
3
ζ
)
ν
2
3
∑
k
=
0
∞
D
k
(
ζ
)
ν
2
k
)
,
asymptotic-expansion
selection
diffop
Hankel-H-1-Bessel-third-kind
𝜈
1
𝜈
𝑧
diffop
Hankel-H-2-Bessel-third-kind
𝜈
1
𝜈
𝑧
4
superscript
𝑒
minus-or-plus
2
𝜋
𝑖
3
𝑧
superscript
1
superscript
𝑧
2
4
𝜁
1
4
superscript
𝑒
minus-or-plus
2
𝜋
𝑖
3
Airy-Ai
superscript
𝑒
plus-or-minus
2
𝜋
𝑖
3
superscript
𝜈
2
3
𝜁
superscript
𝜈
4
3
superscript
subscript
𝑘
0
subscript
𝐶
𝑘
𝜁
superscript
𝜈
2
𝑘
diffop
Airy-Ai
1
superscript
𝑒
plus-or-minus
2
𝜋
𝑖
3
superscript
𝜈
2
3
𝜁
superscript
𝜈
2
3
superscript
subscript
𝑘
0
subscript
𝐷
𝑘
𝜁
superscript
𝜈
2
𝑘
{\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}'\left(\nu z\right)\\
{H^{(2)}_{\nu}}'\left(\nu z\right)}\sim\frac{4e^{\mp 2\pi i/3}}{z}\left(\frac{%
1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}\*\left(\frac{e^{\mp 2\pi i/3}\mathrm{Ai}%
\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{4}{3}}}\sum_{k%
=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{\mathrm{Ai}'\left(e^{\pm 2\pi i%
/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{2}{3}}}\sum_{k=0}^{\infty}\frac{D%
_{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)}}
C
k
(
ζ
)
subscript
𝐶
𝑘
𝜁
{\displaystyle{\displaystyle C_{k}(\zeta)}}
D
k
(
ζ
)
subscript
𝐷
𝑘
𝜁
{\displaystyle{\displaystyle D_{k}(\zeta)}}
Identifiers
