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Statements
I
(
y
,
k
)
=
Δ
1
/
4
π
2
k
1
/
3
exp
(
i
k
f
~
)
(
(
g
+
f
+
′′
+
g
-
-
f
-
′′
)
Ai
(
-
k
2
/
3
Δ
)
(
1
+
O
(
1
k
)
)
-
i
(
g
+
f
+
′′
-
g
-
-
f
-
′′
)
Ai
′
(
-
k
2
/
3
Δ
)
k
1
/
3
Δ
1
/
2
(
1
+
O
(
1
k
)
)
)
,
𝐼
𝑦
𝑘
superscript
Δ
1
4
𝜋
2
superscript
𝑘
1
3
𝑖
𝑘
~
𝑓
subscript
𝑔
superscript
subscript
𝑓
′′
subscript
𝑔
superscript
subscript
𝑓
′′
Airy-Ai
superscript
𝑘
2
3
Δ
1
Big-O
1
𝑘
𝑖
subscript
𝑔
superscript
subscript
𝑓
′′
subscript
𝑔
superscript
subscript
𝑓
′′
diffop
Airy-Ai
1
superscript
𝑘
2
3
Δ
superscript
𝑘
1
3
superscript
Δ
1
2
1
Big-O
1
𝑘
{\displaystyle{\displaystyle I(y,k)=\frac{\Delta^{1/4}\pi\sqrt{2}}{k^{1/3}}%
\exp\left(ik\widetilde{f}\right)\left(\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime%
\prime}}}+\frac{g_{-}}{\sqrt{-f_{-}^{\prime\prime}}}\right)\mathrm{Ai}\left(-k%
^{2/3}\Delta\right)\left(1+O\left(\frac{1}{k}\right)\right)-i\left(\frac{g_{+}%
}{\sqrt{f_{+}^{\prime\prime}}}-\frac{g_{-}}{\sqrt{-f_{-}^{\prime\prime}}}%
\right)\frac{\mathrm{Ai}'\left(-k^{2/3}\Delta\right)}{k^{1/3}\Delta^{1/2}}%
\left(1+O\left(\frac{1}{k}\right)\right)\right),}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
π
{\displaystyle{\displaystyle\pi}}
exp
z
𝑧
{\displaystyle{\displaystyle\exp\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
y
𝑦
{\displaystyle{\displaystyle y}}
k
𝑘
{\displaystyle{\displaystyle k}}
I
(
𝐲
,
k
)
𝐼
𝐲
𝑘
{\displaystyle{\displaystyle I(\mathbf{y},k)}}
f
(
u
,
𝐲
)
𝑓
𝑢
𝐲
{\displaystyle{\displaystyle f(u,\mathbf{y})}}
