Formula:6473

Digital Library of Mathematical Functions ID 18.17.E35

${\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-xz}H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}},}}$

Constraint(s)

${\displaystyle{\displaystyle z\in\mathbb{C}}}$

Symbols List

• ${\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}$: Hermite polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathbb{C}}}$:
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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