DLMF:14.8.E12 (Q4786): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m1kdec

Revision as of 01:34, 2 January 2020

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DLMF:14.8.E12
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    Statements

    Defining formula
    P ν μ ⁡ ( x ) ∼ Γ ⁡ ( ν + 1 2 ) π 1 / 2 ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ( 2 ⁢ x ) ν , asymptotically-equals Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 1 2 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 superscript 2 𝑥 𝜈 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(% \nu+\frac{1}{2}\right)}{\pi^{1/2}\Gamma\left(\nu-\mu+1\right)}(2x)^{\nu},}}
    0 references
    DLMF id
    DLMF:14.8.E12
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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    μ - ν ≠ 1 , 2 , 3 , … 𝜇 𝜈 1 2 3 … {\displaystyle{\displaystyle\mu-\nu\neq 1,2,3,\dots}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1abdec
    0 references
    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1kdec
    0 references
     
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