DLMF:18.17.E28 (Q5769)

From DRMF
Revision as of 15:51, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:18.17.E28
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - x 2 ⁢ 𝐻𝑒 2 ⁢ n + 1 ⁡ ( 2 ⁢ x ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ 1 2 ⁢ π ⁢ e - 1 4 ⁢ y 2 ⁢ 𝐻𝑒 2 ⁢ n + 1 ⁡ ( y ) . superscript subscript 0 superscript 𝑒 superscript 𝑥 2 Hermite-polynomial-He 2 𝑛 1 2 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 1 2 𝜋 superscript 𝑒 1 4 superscript 𝑦 2 Hermite-polynomial-He 2 𝑛 1 𝑦 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n+1}\left% (2x\right)\sin\left(xy\right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-% \frac{1}{4}y^{2}}\mathit{He}_{2n+1}\left(y\right).}}
    0 references
    DLMF id
    DLMF:18.17.E28
    0 references
    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.m2afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aodec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaadec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5769&oldid=61511"