DLMF:14.5.E24 (Q4736)

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DLMF:14.5.E24
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    Statements

    Defining formula
    P 1 2 ⁡ ( cosh ⁡ ξ ) = 2 π ⁢ e ξ / 2 ⁢ E ⁡ ( ( 1 - e - 2 ⁢ ξ ) 1 / 2 ) , shorthand-Legendre-P-first-kind 1 2 𝜉 2 𝜋 superscript 𝑒 𝜉 2 complete-elliptic-integral-second-kind-E superscript 1 superscript 𝑒 2 𝜉 1 2 {\displaystyle{\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi% }e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}\right),}}
    0 references
    DLMF id
    DLMF:14.5.E24
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
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    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2addec
    0 references
    Q11447
    Defining formula
    P ν ⁡ ( z ) = P ν 0 ⁡ ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m3abdec
    0 references
    Q11599
    Defining formula
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S5.XMD1.m1ddec
    0 references
     
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