DLMF:2.8.E35 (Q971): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11090 / rank
 
Normal rank
Property / Symbols used: Q11090 / qualifier
 
test:

env ⁢ J ν ⁡ ( x ) envelope-Bessel-J 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar% {x}\right)}}

\envBesselJ{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Q11090 / qualifier
 
xml-id: C2.S8.SS4.p5.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11085 / rank
 
Normal rank
Property / Symbols used: Q11085 / qualifier
 
test:

W n , j ⁢ ( u , ξ ) subscript 𝑊 𝑛 𝑗 𝑢 𝜉 {\displaystyle{\displaystyle W_{n,j}(u,\xi)}}

W_{n,j}(u,\xi)
Property / Symbols used: Q11085 / qualifier
 
xml-id: C2.S8.XMD20.m1edec
Property / Symbols used
 
Property / Symbols used: Q11086 / rank
 
Normal rank
Property / Symbols used: Q11086 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11086 / qualifier
 
xml-id: C2.S8.XMD21.m1kdec
Property / Symbols used
 
Property / Symbols used: Q11087 / rank
 
Normal rank
Property / Symbols used: Q11087 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11087 / qualifier
 
xml-id: C2.S8.XMD22.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11088 / rank
 
Normal rank
Property / Symbols used: Q11088 / qualifier
 
test:

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

A_{s}(\xi)
Property / Symbols used: Q11088 / qualifier
 
xml-id: C2.S8.XMD23.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11089 / rank
 
Normal rank
Property / Symbols used: Q11089 / qualifier
 
test:

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

B_{s}(\xi)
Property / Symbols used: Q11089 / qualifier
 
xml-id: C2.S8.XMD24.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11069 / rank
 
Normal rank
Property / Symbols used: Q11069 / qualifier
 
test:

u 𝑢 {\displaystyle{\displaystyle u}}

u
Property / Symbols used: Q11069 / qualifier
 
xml-id: C2.S8.XMD4.m1qdec

Latest revision as of 16:30, 1 January 2020

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DLMF:2.8.E35
No description defined

    Statements

    test
    W n , 3 ⁢ ( u , ξ ) = | ξ | 1 / 2 ⁢ J ν ⁡ ( u ⁢ | ξ | 1 / 2 ) ⁢ ∑ s = 0 n - 1 A s ⁢ ( ξ ) u 2 ⁢ s - | ξ | ⁢ J ν + 1 ⁡ ( u ⁢ | ξ | 1 / 2 ) ⁢ ∑ s = 0 n - 2 B s ⁢ ( ξ ) u 2 ⁢ s + 1 + | ξ | 1 / 2 ⁢ env ⁢ J ν ⁡ ( u ⁢ | ξ | 1 / 2 ) ⁢ O ⁡ ( 1 u 2 ⁢ n - 1 ) , subscript 𝑊 𝑛 3 𝑢 𝜉 superscript 𝜉 1 2 Bessel-J 𝜈 𝑢 superscript 𝜉 1 2 superscript subscript 𝑠 0 𝑛 1 subscript 𝐴 𝑠 𝜉 superscript 𝑢 2 𝑠 𝜉 Bessel-J 𝜈 1 𝑢 superscript 𝜉 1 2 superscript subscript 𝑠 0 𝑛 2 subscript 𝐵 𝑠 𝜉 superscript 𝑢 2 𝑠 1 superscript 𝜉 1 2 envelope-Bessel-J 𝜈 𝑢 superscript 𝜉 1 2 Big-O 1 superscript 𝑢 2 𝑛 1 {\displaystyle{\displaystyle W_{n,3}(u,\xi)=|\xi|^{1/2}J_{\nu}\left(u|\xi|^{1/% 2}\right)\sum_{s=0}^{n-1}\frac{A_{s}(\xi)}{u^{2s}}-|\xi|J_{\nu+1}\left(u|\xi|^% {1/2}\right)\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1}}+|\xi|^{1/2}\mathrm{env% }\mskip-2.0mu J_{\nu}\left(u|\xi|^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right)% ,}}
    0 references
    DLMF id
    DLMF:2.8.E35
    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.m2addec
    0 references
    order not exceeding
    test
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2ajdec
    0 references
    Q11090
    test
    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.m2abdec
    0 references
    Q11085
    test
    W n , j ⁢ ( u , ξ ) subscript 𝑊 𝑛 𝑗 𝑢 𝜉 {\displaystyle{\displaystyle W_{n,j}(u,\xi)}}
    xml-id
    C2.S8.XMD20.m1edec
    0 references
    Q11086
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C2.S8.XMD21.m1kdec
    0 references
    Q11087
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S8.XMD22.m1ddec
    0 references
    Q11088
    test
    A s ⁢ ( ξ ) subscript 𝐴 𝑠 𝜉 {\displaystyle{\displaystyle A_{s}(\xi)}}
    xml-id
    C2.S8.XMD23.m1gdec
    0 references
    Q11089
    test
    B s ⁢ ( ξ ) subscript 𝐵 𝑠 𝜉 {\displaystyle{\displaystyle B_{s}(\xi)}}
    xml-id
    C2.S8.XMD24.m1gdec
    0 references
    Q11069
    test
    u 𝑢 {\displaystyle{\displaystyle u}}
    xml-id
    C2.S8.XMD4.m1qdec
    0 references
     
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