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Statements
Ai
(
-
z
)
∼
1
π
z
1
/
4
(
cos
(
ζ
-
1
4
π
)
∑
k
=
0
∞
(
-
1
)
k
u
2
k
ζ
2
k
+
sin
(
ζ
-
1
4
π
)
∑
k
=
0
∞
(
-
1
)
k
u
2
k
+
1
ζ
2
k
+
1
)
,
asymptotic-expansion
Airy-Ai
𝑧
1
𝜋
superscript
𝑧
1
4
𝜁
1
4
𝜋
superscript
subscript
𝑘
0
superscript
1
𝑘
subscript
𝑢
2
𝑘
superscript
𝜁
2
𝑘
𝜁
1
4
𝜋
superscript
subscript
𝑘
0
superscript
1
𝑘
subscript
𝑢
2
𝑘
1
superscript
𝜁
2
𝑘
1
{\displaystyle{\displaystyle\mathrm{Ai}\left(-z\right)\sim\frac{1}{\sqrt{\pi}z%
^{1/4}}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}%
\frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{%
\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),}}
|
ph
z
|
≤
2
3
π
-
δ
phase
𝑧
2
3
𝜋
𝛿
{\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}
Ai
(
z
)
Airy-Ai
𝑧
{\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
z
𝑧
{\displaystyle{\displaystyle z}}
δ
𝛿
{\displaystyle{\displaystyle\delta}}
ζ
𝜁
{\displaystyle{\displaystyle\zeta}}
u
s
subscript
𝑢
𝑠
{\displaystyle{\displaystyle u_{s}}}
