DLMF:14.19.E7 (Q4920)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:14.19.E7
No description defined

    Statements

    Defining formula
    P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) Γ ⁡ ( n - m + 1 2 ) ⁢ ( 2 π ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ 𝑸 m - 1 2 n ⁡ ( coth ⁡ ξ ) , Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript 2 𝜋 𝜉 1 2 associated-Legendre-black-Q 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{% \Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*% \left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}% \left(\coth\xi\right),}}
    0 references
    DLMF id
    DLMF:14.19.E7
    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.m2addec
    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.m2addec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
    0 references
    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2addec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1bdec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1cdec
    0 references
    Q11629
    Defining formula
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD10.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4920&oldid=37178"