DLMF:18.15.E19 (Q5713): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11757 / rank
 
Normal rank
Property / Symbols used: Q11757 / qualifier
 
Defining formula:

A m ⁢ ( ξ ) subscript 𝐴 𝑚 𝜉 {\displaystyle{\displaystyle A_{m}(\xi)}}

A_{m}(\xi)
Property / Symbols used: Q11757 / qualifier
 
xml-id: C18.S15.XMD16.m1dec
Property / Symbols used
 
Property / Symbols used: coefficient (locally) / rank
 
Normal rank
Property / Symbols used: coefficient (locally) / qualifier
 
Defining formula:

B m ⁢ ( ξ ) subscript 𝐵 𝑚 𝜉 {\displaystyle{\displaystyle B_{m}(\xi)}}

B_{m}(\xi)
Property / Symbols used: coefficient (locally) / qualifier
 
xml-id: C18.S15.XMD17.m1dec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1ddec

Latest revision as of 15:39, 2 January 2020

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DLMF:18.15.E19
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    Statements

    Defining formula
    L n ( α ) ⁡ ( ν ⁢ x ) = e 1 2 ⁢ ν ⁢ x 2 α ⁢ x 1 2 ⁢ α + 1 4 ⁢ ( 1 - x ) 1 4 ⁢ ( ξ 1 2 ⁢ J α ⁡ ( ν ⁢ ξ ) ⁢ ∑ m = 0 M - 1 A m ⁢ ( ξ ) ν 2 ⁢ m + ξ - 1 2 ⁢ J α + 1 ⁡ ( ν ⁢ ξ ) ⁢ ∑ m = 0 M - 1 B m ⁢ ( ξ ) ν 2 ⁢ m + 1 + ξ 1 2 ⁢ env ⁢ J α ⁡ ( ν ⁢ ξ ) ⁢ O ⁡ ( 1 ν 2 ⁢ M - 1 ) ) , Laguerre-polynomial-L 𝛼 𝑛 𝜈 𝑥 superscript 𝑒 1 2 𝜈 𝑥 superscript 2 𝛼 superscript 𝑥 1 2 𝛼 1 4 superscript 1 𝑥 1 4 superscript 𝜉 1 2 Bessel-J 𝛼 𝜈 𝜉 superscript subscript 𝑚 0 𝑀 1 subscript 𝐴 𝑚 𝜉 superscript 𝜈 2 𝑚 superscript 𝜉 1 2 Bessel-J 𝛼 1 𝜈 𝜉 superscript subscript 𝑚 0 𝑀 1 subscript 𝐵 𝑚 𝜉 superscript 𝜈 2 𝑚 1 superscript 𝜉 1 2 envelope-Bessel-J 𝛼 𝜈 𝜉 Big-O 1 superscript 𝜈 2 𝑀 1 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=\frac{e^{\frac% {1}{2}\nu x}}{2^{\alpha}x^{\frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}% \left(\xi^{\frac{1}{2}}J_{\alpha}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{A_{m% }(\xi)}{\nu^{2m}}+\xi^{-\frac{1}{2}}J_{\alpha+1}\left(\nu\xi\right)\sum_{m=0}^% {M-1}\frac{B_{m}(\xi)}{\nu^{2m+1}}+\xi^{\frac{1}{2}}\mathrm{env}\mskip-2.0mu J% _{\alpha}\left(\nu\xi\right)O\left(\frac{1}{\nu^{2M-1}}\right)\right),}}
    0 references
    DLMF id
    DLMF:18.15.E19
    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.m2aadec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aedec
    0 references
    Q11333
    Defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2aadec
    0 references
    Q11090
    Defining formula
    env ⁢ J ν ⁡ ( x ) envelope-Bessel-J 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1ldec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ldec
    0 references
    DLMF:18.15.E17
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C18.S15.XMD14.m1adec
    0 references
    DLMF:18.15.E18
    Defining formula
    ξ 𝜉 {\displaystyle{\displaystyle\xi}}
    xml-id
    C18.S15.XMD15.m1adec
    0 references
    Q11757
    Defining formula
    A m ⁢ ( ξ ) subscript 𝐴 𝑚 𝜉 {\displaystyle{\displaystyle A_{m}(\xi)}}
    xml-id
    C18.S15.XMD16.m1dec
    0 references
    coefficient (locally)
    Defining formula
    B m ⁢ ( ξ ) subscript 𝐵 𝑚 𝜉 {\displaystyle{\displaystyle B_{m}(\xi)}}
    xml-id
    C18.S15.XMD17.m1dec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ddec
    0 references
     
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