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Statements
U
(
a
,
x
cos
t
+
y
sin
t
)
=
e
1
4
(
x
sin
t
-
y
cos
t
)
2
∑
m
=
0
∞
(
-
a
-
1
2
m
)
(
tan
t
)
m
U
(
m
+
a
,
x
)
U
(
-
m
-
1
2
,
y
)
,
parabolic-U
𝑎
𝑥
𝑡
𝑦
𝑡
superscript
𝑒
1
4
superscript
𝑥
𝑡
𝑦
𝑡
2
superscript
subscript
𝑚
0
binomial
𝑎
1
2
𝑚
superscript
𝑡
𝑚
parabolic-U
𝑚
𝑎
𝑥
parabolic-U
𝑚
1
2
𝑦
{\displaystyle{\displaystyle U\left(a,x\cos t+y\sin t\right)\\
=e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt%
}{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y%
\right),}}
ℜ
a
≤
-
1
2
,
0
≤
t
≤
1
4
π
formulae-sequence
𝑎
1
2
0
𝑡
1
4
𝜋
{\displaystyle{\displaystyle\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
U
(
a
,
z
)
parabolic-U
𝑎
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
tan
z
𝑧
{\displaystyle{\displaystyle\tan\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
y
𝑦
{\displaystyle{\displaystyle y}}
a
𝑎
{\displaystyle{\displaystyle a}}
