DLMF:14.19.E2 (Q4914): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2aadec

Revision as of 13:29, 2 January 2020

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DLMF:14.19.E2
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    Statements

    Defining formula
    P ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( 1 2 - μ ) π 1 / 2 ⁢ ( 1 - e - 2 ⁢ ξ ) μ ⁢ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 ⁢ μ ; 1 - e - 2 ⁢ ξ ) , Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 𝜋 1 2 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 𝜇 1 2 𝜇 1 superscript 𝑒 2 𝜉 {\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=% \frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{% \mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu% ;1-2\mu;1-e^{-2\xi}\right),}}
    0 references
    DLMF id
    DLMF:14.19.E2
    0 references
    constraint
    μ ≠ 1 2 , 3 2 , 5 2 , … 𝜇 1 2 3 2 5 2 … {\displaystyle{\displaystyle\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots}}
    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.m2adec
    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.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.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
    0 references
     
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