DLMF:18.17.E28 (Q5769): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
(4 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

{\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aaadec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1wdec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1ydec

Latest revision as of 15:51, 2 January 2020

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English
DLMF:18.17.E28
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    Statements

    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:18.17.E28
    0 references
    𝐻𝑒 n ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    C18.S3.T1.t1.r14.m2afdec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2aodec
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1aaadec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2akdec
    0 references
    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3aaadec
    0 references
    sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    C4.S14.E1.m2afdec
    0 references
    y 𝑦 {\displaystyle{\displaystyle y}}
    C18.S1.XMD1.m1wdec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C18.S1.XMD6.m1aadec
    0 references
    x 𝑥 {\displaystyle{\displaystyle x}}
    C18.S2.XMD3.m1ydec
    0 references