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Statements
Ψ
(
H
)
(
0
,
0
,
z
)
=
Ψ
(
H
)
(
0
,
0
,
-
z
)
¯
=
2
1
/
3
3
exp
(
1
27
i
z
3
)
Ψ
(
E
)
(
0
,
0
,
-
z
2
2
/
3
)
,
hyperbolic-umbilic-canonical-integral
0
0
𝑧
hyperbolic-umbilic-canonical-integral
0
0
𝑧
superscript
2
1
3
3
1
27
𝑖
superscript
𝑧
3
elliptic-umbilic-canonical-integral
0
0
𝑧
superscript
2
2
3
{\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(0,0,z\right)=\overline{%
\Psi^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp\left(%
\frac{1}{27}iz^{3}\right)\Psi^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right%
),}}
-
∞
<
z
<
∞
𝑧
{\displaystyle{\displaystyle-\infty<z<\infty}}
z
¯
𝑧
{\displaystyle{\displaystyle\overline{\NVar{z}}}}
Ψ
(
E
)
(
𝐱
)
elliptic-umbilic-canonical-integral
𝐱
{\displaystyle{\displaystyle\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)}}
exp
z
𝑧
{\displaystyle{\displaystyle\exp\NVar{z}}}
Ψ
(
H
)
(
𝐱
)
hyperbolic-umbilic-canonical-integral
𝐱
{\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
z
𝑧
{\displaystyle{\displaystyle z}}
