DLMF:14.15.E24 (Q4871): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11619 / rank
 
Normal rank
Property / Symbols used: Q11619 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q11619 / qualifier
 
xml-id: C14.S15.XMD9.m1gdec

Latest revision as of 01:52, 2 January 2020

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DLMF:14.15.E24
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    Statements

    Defining formula
    𝖯 ν - μ ⁡ ( x ) = 1 ( ν + 1 2 ) 1 / 4 ⁢ 2 ( ν + μ ) / 2 ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 3 4 ) ⁢ ( ζ 2 - α 2 x 2 - a 2 ) 1 / 4 ⁢ ( U ⁡ ( μ - ν - 1 2 , ( 2 ⁢ ν + 1 ) 1 / 2 ⁢ ζ ) + O ⁡ ( ν - 2 / 3 ) ⁢ env ⁢ U ⁡ ( μ - ν - 1 2 , ( 2 ⁢ ν + 1 ) 1 / 2 ⁢ ζ ) ) , Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 1 superscript 𝜈 1 2 1 4 superscript 2 𝜈 𝜇 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 3 4 superscript superscript 𝜁 2 superscript 𝛼 2 superscript 𝑥 2 superscript 𝑎 2 1 4 parabolic-U 𝜇 𝜈 1 2 superscript 2 𝜈 1 1 2 𝜁 Big-O superscript 𝜈 2 3 envelope-Parabolic-U 𝜇 𝜈 1 2 superscript 2 𝜈 1 1 2 𝜁 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{% \left(\nu+\frac{1}{2}\right)^{1/4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+% \frac{1}{2}\mu+\frac{3}{4}\right)}\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2% }}\right)^{1/4}\*\left(U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}% \zeta\right)+O\left(\nu^{-2/3}\right)\mathrm{env}\mskip-1.0mu U\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),}}
    0 references
    DLMF id
    DLMF:14.15.E24
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2andec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2addec
    0 references
    Q11584
    Defining formula
    env ⁢ U ⁡ ( c , x ) envelope-Parabolic-U 𝑐 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-1.0mu U\left(\NVar{c},\NVar{x}% \right)}}
    xml-id
    C14.S15.SS5.p1.m7aadec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1odec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1qdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1qdec
    0 references
    Q11620
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1dec
    0 references
    Q11621
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1dec
    0 references
    Q11619
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1gdec
    0 references
     
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