DLMF:10.23.E8 (Q3462): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2abdec

Revision as of 12:30, 2 January 2020

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DLMF:10.23.E8
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    Statements

    test
    𝒞 ν ⁡ ( w ) w ν = 2 ν ⁢ Γ ⁡ ( ν ) ⁢ ∑ k = 0 ∞ ( ν + k ) ⁢ 𝒞 ν + k ⁡ ( u ) u ν ⁢ J ν + k ⁡ ( v ) v ν ⁢ C k ( ν ) ⁡ ( cos ⁡ α ) , cylinder-function 𝜈 𝑤 superscript 𝑤 𝜈 superscript 2 𝜈 Euler-Gamma 𝜈 superscript subscript 𝑘 0 𝜈 𝑘 cylinder-function 𝜈 𝑘 𝑢 superscript 𝑢 𝜈 Bessel-J 𝜈 𝑘 𝑣 superscript 𝑣 𝜈 ultraspherical-Gegenbauer-polynomial 𝜈 𝑘 𝛼 {\displaystyle{\displaystyle\frac{\mathscr{C}_{\nu}\left(w\right)}{w^{\nu}}=2^% {\nu}\Gamma\left(\nu\right)\*\sum_{k=0}^{\infty}(\nu+k)\frac{\mathscr{C}_{\nu+% k}\left(u\right)}{u^{\nu}}\frac{J_{\nu+k}\left(v\right)}{v^{\nu}}C^{(\nu)}_{k}% \left(\cos\alpha\right),}}
    0 references
    DLMF id
    DLMF:10.23.E8
    0 references
    constraint
    | v ⁢ e ± i ⁢ α | < | u | 𝑣 superscript 𝑒 plus-or-minus 𝑖 𝛼 𝑢 {\displaystyle{\displaystyle|ve^{\pm i\alpha}|<|u|}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aedec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
     
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