DLMF:18.17.E22 (Q5763)

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DLMF:18.17.E22
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ e - 1 4 ⁢ x 2 ⁢ 𝐻𝑒 n ⁡ ( x ) ⁢ e 1 2 ⁢ i ⁢ x ⁢ y ⁢ d x = i n ⁢ e - 1 4 ⁢ y 2 ⁢ 𝐻𝑒 n ⁡ ( y ) , 1 2 𝜋 superscript subscript superscript 𝑒 1 4 superscript 𝑥 2 Hermite-polynomial-He 𝑛 𝑥 superscript 𝑒 1 2 imaginary-unit 𝑥 𝑦 𝑥 imaginary-unit 𝑛 superscript 𝑒 1 4 superscript 𝑦 2 Hermite-polynomial-He 𝑛 𝑦 {\displaystyle{\displaystyle\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-% \frac{1}{4}x^{2}}\mathit{He}_{n}\left(x\right)e^{\frac{1}{2}\mathrm{i}xy}% \mathrm{d}x={\mathrm{i}^{n}}e^{-\frac{1}{4}y^{2}}\mathit{He}_{n}\left(y\right)% ,}}
    0 references
    DLMF id
    DLMF:18.17.E22
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    Symbols used
    Q11525
    Defining formula
    𝐻𝑒 n ⁡ ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r14.m2adec
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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.m2aidec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3audec
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    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1qdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1udec
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