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Statements
Ψ
K
(
𝐱
)
=
2
π
∑
j
=
1
j
max
(
𝐱
)
exp
(
i
(
Φ
K
(
t
j
(
𝐱
)
;
𝐱
)
+
1
4
π
(
-
1
)
j
+
K
+
1
)
)
|
∂
2
Φ
K
(
t
j
(
𝐱
)
;
𝐱
)
∂
t
2
|
-
1
/
2
(
1
+
o
(
1
)
)
.
canonical-integral
𝐾
𝐱
2
𝜋
superscript
subscript
𝑗
1
subscript
𝑗
𝐱
𝑖
cuspoid-catastrophe
𝐾
subscript
𝑡
𝑗
𝐱
𝐱
1
4
𝜋
superscript
1
𝑗
𝐾
1
superscript
partial-derivative
cuspoid-catastrophe
𝐾
subscript
𝑡
𝑗
𝐱
𝐱
𝑡
2
1
2
1
little-o
1
{\displaystyle{\displaystyle\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum%
\limits_{j=1}^{j_{\max}(\mathbf{x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(%
\mathbf{x});\mathbf{x}\right)+\tfrac{1}{4}\pi(-1)^{j+K+1}\right)\right)\left|%
\frac{{\partial}^{2}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)}{{%
\partial t}^{2}}\right|^{-1/2}(1+o\left(1\right)).}}
Ψ
K
(
𝐱
)
canonical-integral
𝐾
𝐱
{\displaystyle{\displaystyle\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)}}
π
{\displaystyle{\displaystyle\pi}}
Φ
K
(
t
;
𝐱
)
cuspoid-catastrophe
𝐾
𝑡
𝐱
{\displaystyle{\displaystyle\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}%
\right)}}
exp
z
𝑧
{\displaystyle{\displaystyle\exp\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
o
(
x
)
little-o
𝑥
{\displaystyle{\displaystyle o\left(\NVar{x}\right)}}
∂
f
∂
x
partial-derivative
𝑓
𝑥
{\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
∂
x
𝑥
{\displaystyle{\displaystyle\partial\NVar{x}}}
t
𝑡
{\displaystyle{\displaystyle t}}
K
𝐾
{\displaystyle{\displaystyle K}}
t
j
(
𝐱
)
subscript
𝑡
𝑗
𝐱
{\displaystyle{\displaystyle t_{j}(\mathbf{x})}}
