DLMF:28.28.E44 (Q8475)

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DLMF:28.28.E44
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    Statements

    test
    1 π 2 ⁢ ∫ 0 2 ⁢ π sin ⁡ ( 2 ⁢ t ) ⁢ se n ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p ⁢ i ⁢ γ ^ n , m ⁢ Dsc 0 ⁡ ( n , m , z ) , 1 superscript 𝜋 2 superscript subscript 0 2 𝜋 2 𝑡 Mathieu-se 𝑛 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 imaginary-unit subscript ^ 𝛾 𝑛 𝑚 Mathieu-Dsc 0 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin\left(% 2t\right)\mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right% )}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{p}\mathrm{i}\widehat{\gamma}_{n,% m}\mathrm{Dsc}_{0}\left(n,m,z\right),}}
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    DLMF id
    DLMF:28.28.E44
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    Symbols used
    Q12319
    test
    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.m1addec
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    Q12283
    test
    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.m7ajdec
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    Q12286
    test
    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.m8akdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aajdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aahdec
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    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2amdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aaddec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aagdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aqdec
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    Q12284
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1ydec
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    Q12294
    test
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1amdec
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    Q12279
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1odec
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    Q12272
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1ahdec
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    DLMF:28.28.E46
    test
    γ ^ n , m subscript ^ 𝛾 𝑛 𝑚 {\displaystyle{\displaystyle\widehat{\gamma}_{n,m}}}
    xml-id
    C28.S28.XMD10.m1dec
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