DLMF:28.28.E42 (Q8473)

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DLMF:28.28.E42
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    Statements

    Defining formula
    sinh ⁡ z π 2 ⁢ ∫ 0 2 ⁢ π cos ⁡ t ⁢ se n ′ ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p ⁢ i ⁢ h ⁢ β ^ n , m ⁢ Dsc 1 ⁡ ( n , m , z ) , 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 𝑡 diffop Mathieu-se 𝑛 1 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 imaginary-unit ℎ subscript ^ 𝛽 𝑛 𝑚 Mathieu-Dsc 1 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t% \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}h\widehat{\beta}_{n,m}% \mathrm{Dsc}_{1}\left(n,m,z\right),}}
    0 references
    DLMF id
    DLMF:28.28.E42
    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.m1abdec
    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.m7ahdec
    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.m8aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aahdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aldec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aafdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aldec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aabdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaedec
    0 references
     
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