DLMF:14.19.E3 (Q4916): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
test:

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

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
test:

𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
xml-id: C15.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
test:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1adec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1adec
Property / Symbols used
 
Property / Symbols used: Q11626 / rank
 
Normal rank
Property / Symbols used: Q11626 / qualifier
 
test:

ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}

\xi>0
Property / Symbols used: Q11626 / qualifier
 
xml-id: C14.S19.XMD7.m1adec

Latest revision as of 12:29, 2 January 2020

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DLMF:14.19.E3
No description defined

    Statements

    test
    𝑸 ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = π 1 / 2 ⁢ ( 1 - e - 2 ⁢ ξ ) μ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( μ + 1 2 , ν + μ + 1 2 ; ν + 1 ; e - 2 ⁢ ξ ) . associated-Legendre-black-Q 𝜇 𝜈 1 2 𝜉 superscript 𝜋 1 2 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 𝜇 1 2 𝜈 𝜇 1 2 𝜈 1 superscript 𝑒 2 𝜉 {\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
    test
    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.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2abdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 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
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1adec
    0 references
    Q11590
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1adec
    0 references
    Q11626
    test
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD7.m1adec
    0 references
     
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