DLMF:18.10.E10 (Q5634)

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DLMF:18.10.E10
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    Statements

    Defining formula
    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.}}
    0 references
    DLMF id
    DLMF:18.10.E10
    0 references
    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.m2abdec
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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.m2ahdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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