DLMF:14.24.E4 (Q4957)

From DRMF
Revision as of 13:41, 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:14.24.E4
No description defined

    Statements

    Defining formula
    𝑸 ν , s μ ⁡ ( z ) = e - s ⁢ μ ⁢ π ⁢ i ⁢ 𝑸 ν μ ⁡ ( z ) - π ⁢ i ⁢ sin ⁡ ( s ⁢ μ ⁢ π ) sin ⁡ ( μ ⁢ π ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ P ν - μ ⁡ ( z ) , associated-Legendre-black-Q 𝜇 𝜈 𝑠 𝑧 superscript 𝑒 𝑠 𝜇 𝜋 𝑖 associated-Legendre-black-Q 𝜇 𝜈 𝑧 𝜋 𝑖 𝑠 𝜇 𝜋 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s% \mu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi% \right)}{\sin\left(\mu\pi\right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}% \left(z\right),}}
    0 references
    DLMF id
    DLMF:14.24.E4
    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.m2aadec
    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
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3abdec
    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11638
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C14.S1.XMD4.m1cdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1cdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1cdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4957&oldid=39019"