DLMF:14.5.E17 (Q4729)

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DLMF:14.5.E17
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    Statements

    Defining formula
    𝑸 Ξ½ Β± 1 / 2 ⁑ ( cosh ⁑ ΞΎ ) = ( Ο€ 2 ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ exp ⁑ ( - ( Ξ½ + 1 2 ) ⁒ ΞΎ ) Ξ“ ⁑ ( Ξ½ + 3 2 ) . associated-Legendre-black-Q plus-or-minus 1 2 𝜈 πœ‰ superscript πœ‹ 2 πœ‰ 1 2 𝜈 1 2 πœ‰ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{\pm 1/2}_{\nu}\left(\cosh\xi\right% )=\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(\nu+\frac{1}{% 2}\right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}.}}
    0 references
    DLMF id
    DLMF:14.5.E17
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q11592
    Defining formula
    𝑸 Ξ½ ΞΌ ⁑ ( z ) associated-Legendre-black-Q πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    exponential function
    Defining formula
    exp ⁑ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁑ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2abdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁑ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2abdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1jdec
    0 references
     
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