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Statements
U
(
-
1
2
μ
2
,
μ
t
2
)
∼
2
g
(
μ
)
(
1
-
t
2
)
1
4
(
cos
κ
∑
s
=
0
∞
(
-
1
)
s
𝒜
~
2
s
(
t
)
μ
4
s
-
sin
κ
∑
s
=
0
∞
(
-
1
)
s
𝒜
~
2
s
+
1
(
t
)
μ
4
s
+
2
)
,
asymptotic-expansion
parabolic-U
1
2
superscript
𝜇
2
𝜇
𝑡
2
2
𝑔
𝜇
superscript
1
superscript
𝑡
2
1
4
𝜅
superscript
subscript
𝑠
0
superscript
1
𝑠
subscript
~
𝒜
2
𝑠
𝑡
superscript
𝜇
4
𝑠
𝜅
superscript
subscript
𝑠
0
superscript
1
𝑠
subscript
~
𝒜
2
𝑠
1
𝑡
superscript
𝜇
4
𝑠
2
{\displaystyle{\displaystyle U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)%
\sim\frac{2g(\mu)}{(1-t^{2})^{\frac{1}{4}}}\*\left(\cos\kappa\sum_{s=0}^{%
\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s}(t)}{\mu^{4s}}-\sin\kappa\sum_{s=%
0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
U
(
a
,
z
)
parabolic-U
𝑎
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
s
𝑠
{\displaystyle{\displaystyle s}}
κ
𝜅
{\displaystyle{\displaystyle\kappa}}
𝒜
~
s
(
t
)
subscript
~
𝒜
𝑠
𝑡
{\displaystyle{\displaystyle\widetilde{\mathcal{A}}_{s}(t)}}
g
(
μ
)
𝑔
𝜇
{\displaystyle{\displaystyle g(\mu)}}
