DLMF:28.28.E28 (Q8455)

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DLMF:28.28.E28
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    Statements

    Defining formula
    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)}.}}
    0 references
    DLMF id
    DLMF:28.28.E28
    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.m1addec
    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.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2awdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aqdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3audec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2afdec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1idec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1wdec
    0 references
    Q12278
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1jdec
    0 references
     
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