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DLMF:28.24.E5 - MaRDI portal

DLMF:28.24.E5 (Q9095)

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DLMF:28.24.E5
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    Statements

    DLMF defining formula
    Ms 2 ⁢ m + 2 ( j ) ⁡ ( z , h ) = ( - 1 ) m ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ B 2 ⁢ ℓ + 2 2 ⁢ m + 2 ⁢ ( h 2 ) B 2 ⁢ s + 2 2 ⁢ m + 2 ⁢ ( h 2 ) ⁢ ( J ℓ - s ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ + s + 2 ( j ) ⁢ ( h ⁢ e z ) - J ℓ + s + 2 ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ - s ( j ) ⁢ ( h ⁢ e z ) ) , modified-Mathieu-Ms 𝑗 2 𝑚 2 𝑧 ℎ superscript 1 𝑚 superscript subscript ℓ 0 superscript 1 ℓ superscript subscript 𝐵 2 ℓ 2 2 𝑚 2 superscript ℎ 2 superscript subscript 𝐵 2 𝑠 2 2 𝑚 2 superscript ℎ 2 Bessel-J ℓ 𝑠 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 2 𝑗 ℎ superscript 𝑒 𝑧 Bessel-J ℓ 𝑠 2 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 𝑗 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle{\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right)=(-1)^{m% }\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^{2% m+2}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+2}^{(j)}(he^{% z})-J_{\ell+s+2}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),}}
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    Symbols used
    Bessel function of the first kind
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2addec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    radial Mathieu function
    DLMF defining formula
    Ms n ( j ) ⁡ ( z , h ) modified-Mathieu-Ms 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E16.m2aadec
    0 references
    integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1cdec
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    parameter
    DLMF defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ddec
    0 references
    integer
    DLMF defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1ddec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1ddec
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    cylinder functions
    DLMF defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1ddec
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    Fourier coefficient
    DLMF defining formula
    B m ⁢ ( q ) subscript 𝐵 𝑚 𝑞 {\displaystyle{\displaystyle B_{m}(q)}}
    xml-id
    C28.S4.XMD2.m1adec
    0 references

    Identifiers

     
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