DLMF:14.23.E6 (Q4952)

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DLMF:14.23.E6
No description defined

    Statements

    Defining formula
    𝖰 ν μ ⁡ ( x ) = e ∓ μ ⁢ π ⁢ i / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ 𝑸 ν μ ⁡ ( x ± i ⁢ 0 ) ± 1 2 ⁢ π ⁢ i ⁢ e ± μ ⁢ π ⁢ i / 2 ⁢ P ν μ ⁡ ( x ± i ⁢ 0 ) . Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 plus-or-minus superscript 𝑒 minus-or-plus 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 associated-Legendre-black-Q 𝜇 𝜈 plus-or-minus 𝑥 𝑖 0 1 2 𝜋 𝑖 superscript 𝑒 plus-or-minus 𝜇 𝜋 𝑖 2 Legendre-P-first-kind 𝜇 𝜈 plus-or-minus 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{\mp\mu\pi i% /2}\Gamma\left(\nu+\mu+1\right)\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)% \pm\tfrac{1}{2}\pi ie^{\pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right).}}
    0 references
    DLMF id
    DLMF:14.23.E6
    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.m2acdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2abdec
    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.m3acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1edec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1edec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1edec
    0 references
     
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