DLMF:18.17.E42 (Q5783)

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DLMF:18.17.E42
No description defined

    Statements

    Defining formula
    ⨍ - 1 1 T n ⁡ ( y ) ⁢ ( 1 - y 2 ) - 1 2 y - x ⁢ d y = π ⁢ U n - 1 ⁡ ( x ) , superscript subscript principal-value-integral 1 1 Chebyshev-polynomial-first-kind-T 𝑛 𝑦 superscript 1 superscript 𝑦 2 1 2 𝑦 𝑥 𝑦 𝜋 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑥 {\displaystyle{\displaystyle\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^% {-\frac{1}{2}}}{y-x}\mathrm{d}y=\pi U_{n-1}\left(x\right),}}
    0 references
    DLMF id
    DLMF:18.17.E42
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    Symbols used
    Q11207
    Defining formula
    T n ⁡ ( x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle T_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r4.m2adec
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    Q11661
    Defining formula
    U n ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle U_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r5.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2atdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaodec
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    Cauchy principal value
    Defining formula
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2adec
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    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1abdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1aodec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1amdec
    0 references
     
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