DLMF:28.23.E9 (Q8393)

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DLMF:28.23.E9
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    Statements

    Defining formula
    Mc 2 ⁢ m + 1 ( j ) ⁡ ( z , h ) = ( - 1 ) m + 1 ⁢ ( ce 2 ⁢ m + 1 ′ ⁡ ( 1 2 ⁢ π , h 2 ) ) - 1 ⁢ coth ⁡ z ⁢ ∑ ℓ = 0 ∞ ( 2 ⁢ ℓ + 1 ) ⁢ A 2 ⁢ ℓ + 1 2 ⁢ m + 1 ⁢ ( h 2 ) ⁢ 𝒞 2 ⁢ ℓ + 1 ( j ) ⁢ ( 2 ⁢ h ⁢ sinh ⁡ z ) , modified-Mathieu-Mc 𝑗 2 𝑚 1 𝑧 ℎ superscript 1 𝑚 1 superscript diffop Mathieu-ce 2 𝑚 1 1 1 2 𝜋 superscript ℎ 2 1 hyperbolic-cotangent 𝑧 superscript subscript ℓ 0 2 ℓ 1 superscript subscript 𝐴 2 ℓ 1 2 𝑚 1 superscript ℎ 2 superscript subscript 𝒞 2 ℓ 1 𝑗 2 ℎ 𝑧 {\displaystyle{\displaystyle{\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m% +1}\left(\mathrm{ce}_{2m+1}'\left(\tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}% \coth z\sum_{\ell=0}^{\infty}(2\ell+1)A_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell% +1}^{(j)}(2h\sinh z),}}
    0 references
    DLMF id
    DLMF:28.23.E9
    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.m7acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2aadec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2acdec
    0 references
    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.m2acdec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1cdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1gdec
    0 references
    Q12291
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1hdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1gdec
    0 references
    Q12304
    Defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1hdec
    0 references
    Q12287
    Defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1cdec
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