DLMF:14.8.E13 (Q4787): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(2 intermediate revisions by the same user not shown)
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.m1ldec
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.m1gdec
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.m1kdec

Latest revision as of 01:34, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.8.E13
No description defined

    Statements

    Defining formula
    P ν μ ⁡ ( x ) ∼ Γ ⁡ ( - ν - 1 2 ) π 1 / 2 ⁢ Γ ⁡ ( - μ - ν ) ⁢ ( 2 ⁢ x ) ν + 1 , asymptotically-equals Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 1 2 superscript 𝜋 1 2 Euler-Gamma 𝜇 𝜈 superscript 2 𝑥 𝜈 1 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(% -\nu-\frac{1}{2}\right)}{\pi^{1/2}\Gamma\left(-\mu-\nu\right)(2x)^{\nu+1}},}}
    0 references
    DLMF id
    DLMF:14.8.E13
    0 references
    constraint
    ℜ ⁡ ν < - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<-\tfrac{1}{2}}}
    0 references
    ℜ ⁡ ν < - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<-\tfrac{1}{2}}}
    0 references
    ν + μ ≠ 0 , 1 , 2 , … 𝜈 𝜇 0 1 2 … {\displaystyle{\displaystyle\nu+\mu\neq 0,1,2,\dots}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
    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.m1acdec
    0 references
    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ldec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1gdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1kdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4787&oldid=32681"