DLMF:28.28.E41 (Q8472)

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DLMF:28.28.E41
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    Statements

    Defining formula
    cosh ⁡ z π 2 ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ se n ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p + 1 ⁢ i ⁢ h ⁢ β ^ n , m ⁢ Dsc 0 ⁡ ( n , m , z ) , 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 𝑡 Mathieu-se 𝑛 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 1 imaginary-unit ℎ subscript ^ 𝛽 𝑛 𝑚 Mathieu-Dsc 0 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin 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+1}\mathrm{i}h\widehat{\beta}_{n,m}% \mathrm{Dsc}_{0}\left(n,m,z\right),}}
    0 references
    DLMF id
    DLMF:28.28.E41
    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.m1aadec
    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.m7agdec
    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.m8ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aagdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaedec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2amdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aaadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaddec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2andec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1vdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ajdec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1ldec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1afdec
    0 references
     
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