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DLMF:28.24.E2 - MaRDI portal

DLMF:28.24.E2 (Q9092)

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DLMF:28.24.E2
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    Statements

    DLMF defining formula
    ε s ⁢ Mc 2 ⁢ m ( j ) ⁡ ( z , h ) = ( - 1 ) m ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ A 2 ⁢ ℓ 2 ⁢ m ⁢ ( h 2 ) A 2 ⁢ s 2 ⁢ m ⁢ ( h 2 ) ⁢ ( J ℓ - s ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ + s ( j ) ⁢ ( h ⁢ e z ) + J ℓ + s ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ - s ( j ) ⁢ ( h ⁢ e z ) ) , subscript 𝜀 𝑠 modified-Mathieu-Mc 𝑗 2 𝑚 𝑧 ℎ superscript 1 𝑚 superscript subscript ℓ 0 superscript 1 ℓ superscript subscript 𝐴 2 ℓ 2 𝑚 superscript ℎ 2 superscript subscript 𝐴 2 𝑠 2 𝑚 superscript ℎ 2 Bessel-J ℓ 𝑠 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 𝑗 ℎ superscript 𝑒 𝑧 Bessel-J ℓ 𝑠 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 𝑗 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle\varepsilon_{s}{\mathrm{Mc}^{(j)}_{2m}}\left(z,h% \right)=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^{2})}{% A_{2s}^{2m}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}^{(j)}% (he^{z})+J_{\ell+s}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),}}
    0 references
    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.m2aadec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    radial Mathieu function
    DLMF 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.m2adec
    0 references
    integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1dec
    0 references
    parameter
    DLMF defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1adec
    0 references
    integer
    DLMF defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1adec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1adec
    0 references
    cylinder functions
    DLMF defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1adec
    0 references
    constants
    DLMF defining formula
    ε s subscript 𝜀 𝑠 {\displaystyle{\displaystyle\varepsilon_{s}}}
    xml-id
    C28.S24.XMD1.m1dec
    0 references
    Fourier coefficient
    DLMF defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1dec
    0 references

    Identifiers

     
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