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Statements
|
Ψ
(
E
)
(
x
,
y
,
z
)
|
2
=
8
π
2
3
2
/
3
∫
0
∞
∫
0
2
π
ℜ
missing
(
Ai
(
1
3
1
/
3
(
x
+
i
y
+
2
z
u
exp
(
i
θ
)
+
3
u
2
exp
(
-
2
i
θ
)
)
)
Bi
(
1
3
1
/
3
(
x
-
i
y
+
2
z
u
exp
(
-
i
θ
)
+
3
u
2
exp
(
2
i
θ
)
)
)
)
u
d
u
d
θ
.
superscript
elliptic-umbilic-canonical-integral
𝑥
𝑦
𝑧
2
8
superscript
𝜋
2
superscript
3
2
3
superscript
subscript
0
superscript
subscript
0
2
𝜋
missing
Airy-Ai
1
superscript
3
1
3
𝑥
𝑖
𝑦
2
𝑧
𝑢
𝑖
𝜃
3
superscript
𝑢
2
2
𝑖
𝜃
Airy-Bi
1
superscript
3
1
3
𝑥
𝑖
𝑦
2
𝑧
𝑢
𝑖
𝜃
3
superscript
𝑢
2
2
𝑖
𝜃
𝑢
𝑢
𝜃
{\displaystyle{\displaystyle\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|%
^{2}=\frac{8\pi^{2}}{3^{2/3}}\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left missing(%
\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+iy+2zu\exp\left(i\theta\right)+3u^{2%
}\exp\left(-2i\theta\right)\right)\right)\*\mathrm{Bi}\left(\frac{1}{3^{1/3}}%
\left(x-iy+2zu\exp\left(-i\theta\right)+3u^{2}\exp\left(2i\theta\right)\right)%
\right)\right)u\mathrm{d}u\mathrm{d}\theta.{}}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
Bi
(
z
)
Airy-Bi
𝑧
{\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
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}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
y
𝑦
{\displaystyle{\displaystyle y}}
z
𝑧
{\displaystyle{\displaystyle z}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
