DLMF:14.19.E3 (Q4916)

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DLMF:14.19.E3
No description defined

    Statements

    Defining formula
    𝑸 Ξ½ - 1 2 ΞΌ ⁑ ( cosh ⁑ ΞΎ ) = Ο€ 1 / 2 ⁒ ( 1 - e - 2 ⁒ ΞΎ ) ΞΌ e ( Ξ½ + ( 1 / 2 ) ) ⁒ ΞΎ ⁒ 𝐅 ⁑ ( ΞΌ + 1 2 , Ξ½ + ΞΌ + 1 2 ; Ξ½ + 1 ; e - 2 ⁒ ΞΎ ) . associated-Legendre-black-Q πœ‡ 𝜈 1 2 πœ‰ superscript πœ‹ 1 2 superscript 1 superscript 𝑒 2 πœ‰ πœ‡ superscript 𝑒 𝜈 1 2 πœ‰ scaled-hypergeometric-bold-F πœ‡ 1 2 𝜈 πœ‡ 1 2 𝜈 1 superscript 𝑒 2 πœ‰ {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh% \xi\right)=\frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}% \*\mathbf{F}\left(\mu+\tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right)% .}}
    0 references
    DLMF id
    DLMF:14.19.E3
    0 references
    Symbols used
    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
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    hyperbolic cosine function
    Defining formula
    cosh ⁑ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2abdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2aadec
    0 references
    Q11591
    Defining formula
    ΞΌ πœ‡ {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1adec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1adec
    0 references
    Q11626
    Defining formula
    ΞΎ > 0 πœ‰ 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD7.m1adec
    0 references
     
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