DLMF:28.28.E46 (Q8477)

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DLMF:28.28.E46
No description defined

    Statements

    Defining formula
    γ ^ n , m = 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π se n ′ ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) ⁢ d t = ( - 1 ) p + 1 ⁢ 4 i ⁢ π ⁢ se n ′ ⁡ ( 0 , h 2 ) ⁢ ce m ⁡ ( 0 , h 2 ) Dsc 1 ⁡ ( n , m , 0 ) . subscript ^ 𝛾 𝑛 𝑚 1 2 𝜋 superscript subscript 0 2 𝜋 diffop Mathieu-se 𝑛 1 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 𝑡 superscript 1 𝑝 1 4 imaginary-unit 𝜋 diffop Mathieu-se 𝑛 1 0 superscript ℎ 2 Mathieu-ce 𝑚 0 superscript ℎ 2 Mathieu-Dsc 1 𝑛 𝑚 0 {\displaystyle{\displaystyle\widehat{\gamma}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2% \pi}\mathrm{se}_{n}'\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)% \mathrm{d}t=(-1)^{p+1}\dfrac{4}{\mathrm{i}\pi}\dfrac{\mathrm{se}_{n}'\left(0,h% ^{2}\right)\mathrm{ce}_{m}\left(0,h^{2}\right)}{\mathrm{Dsc}_{1}\left(n,m,0% \right)}.}}
    0 references
    DLMF id
    DLMF:28.28.E46
    0 references
    Symbols used
    Q12319
    Defining formula
    Dsc j ⁡ ( n , m , z ) Mathieu-Dsc 𝑗 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},% \NVar{z}\right)}}
    xml-id
    C28.S28.E40.m1afdec
    0 references
    Q12283
    Defining formula
    ce n ⁡ ( z , q ) Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m7aldec
    0 references
    Q12286
    Defining formula
    se n ⁡ ( z , q ) Mathieu-se 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m8amdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aaldec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aajdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aafdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaidec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1aadec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1aodec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1qdec
    0 references
     
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