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Statements
Ψ
(
H
)
(
𝐱
)
=
4
π
/
6
exp
(
i
(
1
27
z
3
+
1
6
z
(
y
+
x
)
+
1
4
π
)
)
∫
∞
exp
(
5
π
i
/
12
)
∞
exp
(
π
i
/
12
)
exp
(
i
(
2
u
6
+
2
z
u
4
+
(
1
2
z
2
+
x
+
y
)
u
2
-
(
y
-
x
)
2
24
u
2
)
)
d
u
,
hyperbolic-umbilic-canonical-integral
𝐱
4
𝜋
6
𝑖
1
27
superscript
𝑧
3
1
6
𝑧
𝑦
𝑥
1
4
𝜋
superscript
subscript
5
𝜋
𝑖
12
𝜋
𝑖
12
𝑖
2
superscript
𝑢
6
2
𝑧
superscript
𝑢
4
1
2
superscript
𝑧
2
𝑥
𝑦
superscript
𝑢
2
superscript
𝑦
𝑥
2
24
superscript
𝑢
2
𝑢
{\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{%
\ifrac{\pi}{6}}\,\exp\left(i\left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac%
{1}{4}\pi\right)\right)\*\int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp%
\left(\pi i/12\right)}\exp\left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+%
x+y\right)u^{2}-\frac{(y-x)^{2}}{24u^{2}}\right)\right)\mathrm{d}u,}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
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}}}
∫
{\displaystyle{\displaystyle\int}}
y
𝑦
{\displaystyle{\displaystyle y}}
z
𝑧
{\displaystyle{\displaystyle z}}
x
𝑥
{\displaystyle{\displaystyle x}}
