DLMF:18.17.E48 (Q5789): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2audec
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.m3aasdec
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.m1addec
Property / Symbols used
 
Property / Symbols used: Q11730 / rank
 
Normal rank
Property / Symbols used: Q11730 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11730 / qualifier
 
xml-id: C18.S1.XMD5.m1edec
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.m1audec
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.m1asdec

Latest revision as of 01:20, 2 January 2020

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English
DLMF:18.17.E48
No description defined

    Statements

    Defining formula
    ∫ - ∞ ∞ H m ⁡ ( y ) ⁢ e - y 2 ⁢ H n ⁡ ( x - y ) ⁢ e - ( x - y ) 2 ⁢ d y = π 1 2 ⁢ 2 - 1 2 ⁢ ( m + n + 1 ) ⁢ H m + n ⁡ ( 2 - 1 2 ⁢ x ) ⁢ e - 1 2 ⁢ x 2 . superscript subscript Hermite-polynomial-H 𝑚 𝑦 superscript 𝑒 superscript 𝑦 2 Hermite-polynomial-H 𝑛 𝑥 𝑦 superscript 𝑒 superscript 𝑥 𝑦 2 𝑦 superscript 𝜋 1 2 superscript 2 1 2 𝑚 𝑛 1 Hermite-polynomial-H 𝑚 𝑛 superscript 2 1 2 𝑥 superscript 𝑒 1 2 superscript 𝑥 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{m}\left(y\right)e^{-y^{2% }}H_{n}\left(x-y\right)e^{-(x-y)^{2}}\mathrm{d}y=\pi^{\frac{1}{2}}2^{-\frac{1}% {2}(m+n+1)}H_{m+n}\left(2^{-\frac{1}{2}}x\right)e^{-\frac{1}{2}x^{2}}.}}
    0 references
    DLMF id
    DLMF:18.17.E48
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2avdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaudec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2audec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aasdec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1addec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1audec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1asdec
    0 references
     
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