DLMF:28.24.E6 (Q8403)

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DLMF:28.24.E6
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    Statements

    Defining formula
    ε s ⁢ Ie 2 ⁢ m ⁡ ( z , h ) = ( - 1 ) s ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ A 2 ⁢ ℓ 2 ⁢ m ⁢ ( h 2 ) A 2 ⁢ s 2 ⁢ m ⁢ ( h 2 ) ⁢ ( I ℓ - s ⁡ ( h ⁢ e - z ) ⁢ I ℓ + s ⁡ ( h ⁢ e z ) + I ℓ + s ⁡ ( h ⁢ e - z ) ⁢ I ℓ - s ⁡ ( h ⁢ e z ) ) , subscript 𝜀 𝑠 modified-Mathieu-Ie 2 𝑚 𝑧 ℎ superscript 1 𝑠 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 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle\varepsilon_{s}\mathrm{Ie}_{2m}\left(z,h\right)=(-% 1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2% m}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s}\left(he^{z}\right)+I_% {\ell+s}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}\right)\right),}}
    0 references
    DLMF id
    DLMF:28.24.E6
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    Symbols used
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    modified Mathieu function
    Defining formula
    Ie n ⁡ ( z , h ) modified-Mathieu-Ie 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\mathrm{Ie}_{\NVar{n}}\left(\NVar{z},\NVar{h}% \right)}}
    xml-id
    C28.S20.E17.m2adec
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    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1ddec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1edec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1edec
    0 references
    Q12305
    Defining formula
    ε s subscript 𝜀 𝑠 {\displaystyle{\displaystyle\varepsilon_{s}}}
    xml-id
    C28.S24.XMD1.m1adec
    0 references
     
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