DLMF:10.71.E3 (Q3904)

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DLMF:10.71.E3
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    Statements

    Defining formula
    ∫ x ⁢ ( f ν ⁢ g ^ ν - g ν ⁢ f ^ ν ) ⁢ d x = x 2 ⁢ 2 ⁢ ( f ^ ν ⁢ ( f ν + 1 + g ν + 1 ) - g ^ ν ⁢ ( f ν + 1 - g ν + 1 ) - f ν ⁢ ( f ^ ν + 1 + g ^ ν + 1 ) + g ν ⁢ ( f ^ ν + 1 - g ^ ν + 1 ) ) = 1 2 ⁢ x ⁢ ( f ν ′ ⁢ f ^ ν - f ν ⁢ f ^ ν ′ + g ν ′ ⁢ g ^ ν - g ν ⁢ g ^ ν ′ ) , 𝑥 subscript 𝑓 𝜈 subscript ^ 𝑔 𝜈 subscript 𝑔 𝜈 subscript ^ 𝑓 𝜈 𝑥 𝑥 2 2 subscript ^ 𝑓 𝜈 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 subscript ^ 𝑔 𝜈 subscript 𝑓 𝜈 1 subscript 𝑔 𝜈 1 subscript 𝑓 𝜈 subscript ^ 𝑓 𝜈 1 subscript ^ 𝑔 𝜈 1 subscript 𝑔 𝜈 subscript ^ 𝑓 𝜈 1 subscript ^ 𝑔 𝜈 1 1 2 𝑥 superscript subscript 𝑓 𝜈 ′ subscript ^ 𝑓 𝜈 subscript 𝑓 𝜈 superscript subscript ^ 𝑓 𝜈 ′ superscript subscript 𝑔 𝜈 ′ subscript ^ 𝑔 𝜈 subscript 𝑔 𝜈 subscript superscript ^ 𝑔 ′ 𝜈 {\displaystyle{\displaystyle\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}% _{\nu})\mathrm{d}x=\frac{x}{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu% +1})-\widehat{g}_{\nu}(f_{\nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+% \widehat{g}_{\nu+1})+g_{\nu}(\widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)=% \tfrac{1}{2}x(f_{\nu}^{\prime}\widehat{f}_{\nu}-f_{\nu}\widehat{f}_{\nu}^{% \prime}+g_{\nu}^{\prime}\widehat{g}_{\nu}-g_{\nu}\widehat{g}^{\prime}_{\nu}),}}
    0 references
    DLMF id
    DLMF:10.71.E3
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1bdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1bdec
    0 references
    Q11487
    Defining formula
    f ν subscript 𝑓 𝜈 {\displaystyle{\displaystyle f_{\nu}}}
    xml-id
    C10.S63.XMD1.m1bdec
    0 references
    Q11488
    Defining formula
    g ν subscript 𝑔 𝜈 {\displaystyle{\displaystyle g_{\nu}}}
    xml-id
    C10.S63.XMD2.m1bdec
    0 references
    Q11489
    Defining formula
    f ^ ν subscript ^ 𝑓 𝜈 {\displaystyle{\displaystyle\widehat{f}_{\nu}}}
    xml-id
    C10.S71.XMD1.m1dec
    0 references
    Q11490
    Defining formula
    g ^ ν subscript ^ 𝑔 𝜈 {\displaystyle{\displaystyle\widehat{g}_{\nu}}}
    xml-id
    C10.S71.XMD2.m1dec
    0 references
     
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