DLMF:14.15.E30 (Q4878): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
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)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2addec
Property / Symbols used
 
Property / Symbols used: Q11320 / rank
 
Normal rank
Property / Symbols used: Q11320 / qualifier
 
Defining formula:

U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}

\paraU@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11320 / qualifier
 
xml-id: C12.S2.SS1.p1.m1abdec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1tdec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1tdec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1tdec
Property / Symbols used
 
Property / Symbols used: Q11620 / rank
 
Normal rank
Property / Symbols used: Q11620 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11620 / qualifier
 
xml-id: C14.S15.XMD13.m1edec
Property / Symbols used
 
Property / Symbols used: Q11621 / rank
 
Normal rank
Property / Symbols used: Q11621 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11621 / qualifier
 
xml-id: C14.S15.XMD14.m1edec
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.m1ldec

Latest revision as of 13:21, 2 January 2020

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English
DLMF:14.15.E30
No description defined

    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 ) ⁢ ( 1 + O ⁡ ( ν - 1 ⁢ ln ⁡ ν ) ) , 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 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 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2apdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( 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 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.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 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1abdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1tdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\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 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1edec
    0 references
    Q11621
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1edec
    0 references
    Q11619
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1ldec
    0 references
     
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