DLMF:18.17.E35 (Q5776): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10923 / rank
 
Normal rank
Property / Symbols used: Q10923 / qualifier
 
Defining formula:

ℂ {\displaystyle{\displaystyle\mathbb{C}}}

\Complexes
Property / Symbols used: Q10923 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r1.m2aadec

Revision as of 01:13, 2 January 2020

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DLMF:18.17.E35
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    Statements

    Defining formula
    ∫ - ∞ ∞ e - x ⁢ z ⁢ H n ⁡ ( x ) ⁢ e - x 2 ⁢ d x = π 1 2 ⁢ ( - z ) n ⁢ e 1 4 ⁢ z 2 , superscript subscript superscript 𝑒 𝑥 𝑧 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑒 superscript 𝑥 2 𝑥 superscript 𝜋 1 2 superscript 𝑧 𝑛 superscript 𝑒 1 4 superscript 𝑧 2 {\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}},}}
    0 references
    DLMF id
    DLMF:18.17.E35
    0 references
    constraint
    z ∈ ℂ 𝑧 {\displaystyle{\displaystyle z\in\mathbb{C}}}
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    Symbols used
    Q10922
    Defining formula
    H n ⁡ ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r13.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ardec
    0 references
    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2aadec
    0 references
     
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