DLMF:28.28.E26 (Q8453)

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DLMF:28.28.E26
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    Statements

    Defining formula
    cosh ⁡ z π 2 ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ me ν ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) m + 1 ⁢ i ⁢ h ⁢ α ν , m ( 1 ) ⁢ D 0 ⁡ ( ν , ν + 2 ⁢ m + 1 , z ) , 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 𝑡 Mathieu-me 𝜈 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑚 1 imaginary-unit ℎ subscript superscript 𝛼 1 𝜈 𝑚 Mathieu-D 0 𝜈 𝜈 2 𝑚 1 𝑧 {\displaystyle{\displaystyle\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t% \mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}% \right)}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(1)% }_{\nu,m}\mathrm{D}_{0}\left(\nu,\nu+2m+1,z\right),}}
    0 references
    DLMF id
    DLMF:28.28.E26
    0 references
    Symbols used
    Q12317
    Defining formula
    D j ⁡ ( ν , μ , z ) Mathieu-D 𝑗 𝜈 𝜇 𝑧 {\displaystyle{\displaystyle\mathrm{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},% \NVar{z}\right)}}
    xml-id
    C28.S28.E24.m1abdec
    0 references
    Q12299
    Defining formula
    me n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S12.SS2.p2.m2aedec
    0 references
     
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