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Statements
Ai
(
-
z
)
=
(
z
/
3
)
(
J
1
/
3
(
ζ
)
+
J
-
1
/
3
(
ζ
)
)
=
1
2
z
/
3
(
e
π
i
/
6
H
1
/
3
(
1
)
(
ζ
)
+
e
-
π
i
/
6
H
1
/
3
(
2
)
(
ζ
)
)
=
1
2
z
/
3
(
e
-
π
i
/
6
H
-
1
/
3
(
1
)
(
ζ
)
+
e
π
i
/
6
H
-
1
/
3
(
2
)
(
ζ
)
)
,
Airy-Ai
𝑧
𝑧
3
Bessel-J
1
3
𝜁
Bessel-J
1
3
𝜁
1
2
𝑧
3
superscript
𝑒
𝜋
𝑖
6
Hankel-H-1-Bessel-third-kind
1
3
𝜁
superscript
𝑒
𝜋
𝑖
6
Hankel-H-2-Bessel-third-kind
1
3
𝜁
1
2
𝑧
3
superscript
𝑒
𝜋
𝑖
6
Hankel-H-1-Bessel-third-kind
1
3
𝜁
superscript
𝑒
𝜋
𝑖
6
Hankel-H-2-Bessel-third-kind
1
3
𝜁
{\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/%
3}\left(\zeta\right)+J_{-1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}%
\left(e^{\pi i/6}{H^{(1)}_{1/3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}%
\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1%
/3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\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)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ζ
𝜁
{\displaystyle{\displaystyle\zeta}}
