DLMF:28.28.E20 (Q8446)

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DLMF:28.28.E20
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    Statements

    Defining formula
    2 π ⁢ ∫ 0 π 𝒞 2 ⁢ ℓ ( j ) ⁢ ( 2 ⁢ h ⁢ R ) ⁢ cos ⁡ ( 2 ⁢ ℓ ⁢ ϕ ) ⁢ ce 2 ⁢ m ⁡ ( t , h 2 ) ⁢ d t = ε ℓ ⁢ ( - 1 ) ℓ + m ⁢ A 2 ⁢ ℓ 2 ⁢ m ⁢ ( h 2 ) ⁢ Mc 2 ⁢ m ( j ) ⁡ ( z , h ) , 2 𝜋 superscript subscript 0 𝜋 subscript superscript 𝒞 𝑗 2 ℓ 2 ℎ 𝑅 2 ℓ italic-ϕ Mathieu-ce 2 𝑚 𝑡 superscript ℎ 2 𝑡 subscript 𝜀 ℓ superscript 1 ℓ 𝑚 subscript superscript 𝐴 2 𝑚 2 ℓ superscript ℎ 2 modified-Mathieu-Mc 𝑗 2 𝑚 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2% \ell}(2hR)\cos\left(2\ell\phi\right)\mathrm{ce}_{2m}\left(t,h^{2}\right)% \mathrm{d}t=\varepsilon_{\ell}(-1)^{\ell+m}A^{2m}_{2\ell}(h^{2}){\mathrm{Mc}^{% (j)}_{2m}}\left(z,h\right),}}
    0 references
    DLMF id
    DLMF:28.28.E20
    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.m7aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
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    radial Mathieu function
    Defining formula
    Mc n ( j ) ⁡ ( z , h ) modified-Mathieu-Mc 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E15.m2addec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1bdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1odec
    0 references
    Q12291
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1ddec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1pdec
    0 references
    Q12304
    Defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1adec
    0 references
    Q12314
    Defining formula
    ϕ ⁢ ( z , t ) italic-ϕ 𝑧 𝑡 {\displaystyle{\displaystyle\phi(z,t)}}
    xml-id
    C28.S28.XMD4.m1bdec
    0 references
    Q12315
    Defining formula
    R ⁢ ( z , t ) 𝑅 𝑧 𝑡 {\displaystyle{\displaystyle R(z,t)}}
    xml-id
    C28.S28.XMD5.m1bdec
    0 references
    Q12316
    Defining formula
    ε ℓ subscript 𝜀 ℓ {\displaystyle{\displaystyle\varepsilon_{\ell}}}
    xml-id
    C28.S28.XMD6.m1dec
    0 references
    Q12287
    Defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1adec
    0 references
     
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