DLMF:18.17.E43 (Q5784): Difference between revisions

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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.m1acdec

Revision as of 01:19, 2 January 2020

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DLMF:18.17.E43
No description defined

    Statements

    Defining formula
    ⨍ - 1 1 U n - 1 ⁡ ( y ) ⁢ ( 1 - y 2 ) 1 2 y - x ⁢ d y = - π ⁢ T n ⁡ ( x ) . superscript subscript principal-value-integral 1 1 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑦 superscript 1 superscript 𝑦 2 1 2 𝑦 𝑥 𝑦 𝜋 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2}% )^{\frac{1}{2}}}{y-x}\mathrm{d}y=-\pi T_{n}\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.17.E43
    0 references
    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.m2aadec
    0 references
    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.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2audec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aapdec
    0 references
    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.m2aadec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1acdec
    0 references
     
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