DLMF:28.28.E28 (Q8455)

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DLMF:28.28.E28
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    α ν , m ( 1 ) = 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ me ν ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) ⁢ d t = ( - 1 ) m + 1 ⁢ 2 ⁢ i π ⁢ me ν ′ ⁡ ( 0 , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( 0 , h 2 ) h ⁢ D 1 ⁡ ( ν , ν + 2 ⁢ m + 1 , 0 ) . subscript superscript 𝛼 1 𝜈 𝑚 1 2 𝜋 superscript subscript 0 2 𝜋 𝑡 Mathieu-me 𝜈 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 𝑡 superscript 1 𝑚 1 2 imaginary-unit 𝜋 diffop Mathieu-me 𝜈 1 0 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 0 superscript ℎ 2 ℎ Mathieu-D 1 𝜈 𝜈 2 𝑚 1 0 {\displaystyle{\displaystyle\alpha^{(1)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi% }\sin t\mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{% 2}\right)\mathrm{d}t=(-1)^{m+1}\dfrac{2\mathrm{i}}{\pi}\dfrac{\mathrm{me}_{\nu% }'\left(0,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(0,h^{2}\right)}{h\mathrm{D}% _{1}\left(\nu,\nu+2m+1,0\right)}.}}
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    DLMF id
    DLMF:28.28.E28
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