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Statements
H
n
(
x
)
=
(
-
2
i
)
n
e
x
2
π
1
2
∫
-
∞
∞
e
-
t
2
t
n
e
2
i
x
t
d
t
=
2
n
+
1
π
1
2
e
x
2
∫
0
∞
e
-
t
2
t
n
cos
(
2
x
t
-
1
2
n
π
)
d
t
.
Hermite-polynomial-H
𝑛
𝑥
superscript
2
𝑖
𝑛
superscript
𝑒
superscript
𝑥
2
superscript
𝜋
1
2
superscript
subscript
superscript
𝑒
superscript
𝑡
2
superscript
𝑡
𝑛
superscript
𝑒
2
𝑖
𝑥
𝑡
𝑡
superscript
2
𝑛
1
superscript
𝜋
1
2
superscript
𝑒
superscript
𝑥
2
superscript
subscript
0
superscript
𝑒
superscript
𝑡
2
superscript
𝑡
𝑛
2
𝑥
𝑡
1
2
𝑛
𝜋
𝑡
{\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{(-2i)^{n}e^{x^{2}}}{\pi%
^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac%
{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left%
(2xt-\tfrac{1}{2}n\pi\right)\mathrm{d}t.}}
H
n
(
x
)
Hermite-polynomial-H
𝑛
𝑥
{\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
