DLMF:28.24.E3 (Q8400): Difference between revisions

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‎Changed an Item: Add constraint
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Property / Symbols used
 
Property / Symbols used: Q12284 / rank
 
Normal rank
Property / Symbols used: Q12284 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12284 / qualifier
 
xml-id: C28.S1.XMD1.m1adec

Revision as of 15:11, 2 January 2020

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DLMF:28.24.E3
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    Statements

    Defining formula
    Mc 2 ⁢ m + 1 ( j ) ⁡ ( z , h ) = ( - 1 ) m ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ A 2 ⁢ ℓ + 1 2 ⁢ m + 1 ⁢ ( h 2 ) A 2 ⁢ s + 1 2 ⁢ m + 1 ⁢ ( h 2 ) ⁢ ( J ℓ - s ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ + s + 1 ( j ) ⁢ ( h ⁢ e z ) + J ℓ + s + 1 ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ - s ( j ) ⁢ ( h ⁢ e z ) ) , modified-Mathieu-Mc 𝑗 2 𝑚 1 𝑧 ℎ superscript 1 𝑚 superscript subscript ℓ 0 superscript 1 ℓ superscript subscript 𝐴 2 ℓ 1 2 𝑚 1 superscript ℎ 2 superscript subscript 𝐴 2 𝑠 1 2 𝑚 1 superscript ℎ 2 Bessel-J ℓ 𝑠 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 1 𝑗 ℎ superscript 𝑒 𝑧 Bessel-J ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 𝑗 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle{\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m% }\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2% m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{% z})+J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),}}
    0 references
    DLMF id
    DLMF:28.24.E3
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    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.m2aadec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1adec
    0 references
     
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