DLMF:18.11.E5 (Q5639): Difference between revisions

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

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

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec

Revision as of 15:29, 2 January 2020

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DLMF:18.11.E5
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    Statements

    Defining formula
    lim n → ∞ ⁡ 1 n α ⁢ P n ( α , β ) ⁡ ( 1 - z 2 2 ⁢ n 2 ) = lim n → ∞ ⁡ 1 n α ⁢ P n ( α , β ) ⁡ ( cos ⁡ z n ) = 2 α z α ⁢ J α ⁡ ( z ) . subscript → 𝑛 1 superscript 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 𝑧 2 2 superscript 𝑛 2 subscript → 𝑛 1 superscript 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑧 𝑛 superscript 2 𝛼 superscript 𝑧 𝛼 Bessel-J 𝛼 𝑧 {\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,% \beta)}_{n}\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{% \alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z% ^{\alpha}}J_{\alpha}\left(z\right).}}
    0 references
    DLMF id
    DLMF:18.11.E5
    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.m2adec
    0 references
    Q11660
    Defining formula
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
     
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