DLMF:18.17.E35 (Q5776)

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DLMF:18.17.E35
No description defined

    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}},}}
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    DLMF id
    DLMF:18.17.E35
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    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
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ardec
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    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aahdec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ardec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aahdec
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    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1bdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ahdec
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    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1afdec
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