DLMF:28.28.E34 (Q8461)

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DLMF:28.28.E34
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    Defining formula
    1 π 2 ⁢ ⨍ 0 2 ⁢ π me ν ′ ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) sin ⁡ t ⁢ d t = ( - 1 ) m + 1 ⁢ i ⁢ h ⁢ α ν , m ( 0 ) ⁢ D 1 ⁡ ( ν , ν + 2 ⁢ m + 1 , 0 ) , 1 superscript 𝜋 2 superscript subscript principal-value-integral 0 2 𝜋 diffop Mathieu-me 𝜈 1 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 𝑡 𝑡 superscript 1 𝑚 1 imaginary-unit ℎ subscript superscript 𝛼 0 𝜈 𝑚 Mathieu-D 1 𝜈 𝜈 2 𝑚 1 0 {\displaystyle{\displaystyle\dfrac{1}{\pi^{2}}\pvint_{0}^{2\pi}\dfrac{\mathrm{% me}_{\nu}'\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{% \sin t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\mathrm{D}_{1}% \left(\nu,\nu+2m+1,0\right),}}
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    DLMF id
    DLMF:28.28.E34
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