DLMF:14.12.E7 (Q4828)

From DRMF
Revision as of 01:43, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:14.12.E7
No description defined

    Statements

    Defining formula
    P ν m ⁡ ( x ) = ( ν + 1 ) m π ⁢ ∫ 0 π ( x + ( x 2 - 1 ) 1 / 2 ⁢ cos ⁡ ϕ ) ν ⁢ cos ⁡ ( m ⁢ ϕ ) ⁢ d ϕ , Legendre-P-first-kind 𝑚 𝜈 𝑥 Pochhammer 𝜈 1 𝑚 𝜋 superscript subscript 0 𝜋 superscript 𝑥 superscript superscript 𝑥 2 1 1 2 italic-ϕ 𝜈 𝑚 italic-ϕ italic-ϕ {\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{{\left(\nu+1% \right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi% \right)^{\nu}\cos\left(m\phi\right)\mathrm{d}\phi,}}
    0 references
    DLMF id
    DLMF:14.12.E7
    0 references
    Symbols used
    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
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ddec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4828&oldid=34068"