DLMF:28.31.E12 (Q8529)

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DLMF:28.31.E12
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    Statements

    Defining formula
    1 π ⁢ ∫ 0 2 ⁢ π ( C p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 π ⁢ ∫ 0 2 ⁢ π ( S p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 , 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝐶 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝑆 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi% )\right)^{2}\mathrm{d}x=\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)% \right)^{2}\mathrm{d}x=1,}}
    0 references
    DLMF id
    DLMF:28.31.E12
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1bdec
    0 references
    Q12285
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C28.S1.XMD4.m1adec
    0 references
    Q12327
    Defining formula
    ξ 𝜉 {\displaystyle{\displaystyle\xi}}
    xml-id
    C28.S31.XMD5.m1hdec
    0 references
    Ince polynomials (locally)
    Defining formula
    C p m ⁢ ( z , ξ ) superscript subscript 𝐶 𝑝 𝑚 𝑧 𝜉 {\displaystyle{\displaystyle C_{p}^{m}(z,\xi)}}
    xml-id
    C28.S31.XMD6.m1adec
    0 references
    Ince polynomials (locally)
    Defining formula
    S p m ⁢ ( z , N ⁢ V ⁢ a ⁢ r ⁢ x ⁢ i ) superscript subscript 𝑆 𝑝 𝑚 𝑧 𝑁 𝑉 𝑎 𝑟 𝑥 𝑖 {\displaystyle{\displaystyle S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\ NVar{xi})}}
    xml-id
    C28.S31.XMD7.m1adec
    0 references
     
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