DLMF:14.15.E30 (Q4878)

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DLMF:14.15.E30
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    Statements

    Defining formula
    𝖯 Ξ½ - ΞΌ ⁑ ( x ) = 1 ( Ξ½ + 1 2 ) 1 / 4 ⁒ 2 ( Ξ½ + ΞΌ ) / 2 ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 3 4 ) ⁒ ( ΞΆ 2 + Ξ± 2 x 2 + a 2 ) 1 / 4 ⁒ U ⁑ ( ΞΌ - Ξ½ - 1 2 , ( 2 ⁒ Ξ½ + 1 ) 1 / 2 ⁒ ΞΆ ) ⁒ ( 1 + O ⁑ ( Ξ½ - 1 ⁒ ln ⁑ Ξ½ ) ) , Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 1 superscript 𝜈 1 2 1 4 superscript 2 𝜈 πœ‡ 2 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 3 4 superscript superscript 𝜁 2 superscript 𝛼 2 superscript π‘₯ 2 superscript π‘Ž 2 1 4 parabolic-U πœ‡ 𝜈 1 2 superscript 2 𝜈 1 1 2 𝜁 1 Big-O superscript 𝜈 1 𝜈 {\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}\*U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta% \right)\left(1+O\left(\nu^{-1}\ln\nu\right)\right),}}
    0 references
    DLMF id
    DLMF:14.15.E30
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁑ ( x ) Big-O π‘₯ {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2apdec
    0 references
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 Ξ½ ΞΌ ⁑ ( x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aedec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    Q11320
    Defining formula
    U ⁑ ( a , z ) parabolic-U π‘Ž 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1abdec
    0 references
    Q11589
    Defining formula
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1tdec
    0 references
    Q11591
    Defining formula
    ΞΌ πœ‡ {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1tdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1tdec
    0 references
    Q11620
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1edec
    0 references
    Q11621
    Defining formula
    ΢ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1edec
    0 references
    Q11619
    Defining formula
    Ξ± 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1ldec
    0 references
     
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