DLMF:28.10.E4 (Q8283)

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DLMF:28.10.E4
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    Statements

    Defining formula
    2 π ⁢ ∫ 0 π / 2 cos ⁡ z ⁢ cos ⁡ t ⁢ cosh ⁡ ( 2 ⁢ h ⁢ sin ⁡ z ⁢ sin ⁡ t ) ⁢ ce 2 ⁢ n + 1 ⁡ ( t , h 2 ) ⁢ d t = A 1 2 ⁢ n + 1 ⁢ ( h 2 ) 2 ⁢ ce 2 ⁢ n + 1 ⁡ ( 0 , h 2 ) ⁢ ce 2 ⁢ n + 1 ⁡ ( z , h 2 ) , 2 𝜋 superscript subscript 0 𝜋 2 𝑧 𝑡 2 ℎ 𝑧 𝑡 Mathieu-ce 2 𝑛 1 𝑡 superscript ℎ 2 𝑡 superscript subscript 𝐴 1 2 𝑛 1 superscript ℎ 2 2 Mathieu-ce 2 𝑛 1 0 superscript ℎ 2 Mathieu-ce 2 𝑛 1 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t% \cosh\left(2h\sin z\sin t\right)\mathrm{ce}_{2n+1}\left(t,h^{2}\right)\mathrm{% d}t=\frac{A_{1}^{2n+1}(h^{2})}{2\mathrm{ce}_{2n+1}\left(0,h^{2}\right)}\mathrm% {ce}_{2n+1}\left(z,h^{2}\right),}}
    0 references
    DLMF id
    DLMF:28.10.E4
    0 references
    Symbols used
    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.m7acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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