DLMF:14.25.E2 (Q4959): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11592 / rank
 
Normal rank
Property / Symbols used: Q11592 / qualifier
 
test:

𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}

\assLegendreOlverQ[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11592 / qualifier
 
xml-id: C14.S21.SS1.p1.m3adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aadec
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.m2aadec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
test:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aadec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11638 / rank
 
Normal rank
Property / Symbols used: Q11638 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11638 / qualifier
 
xml-id: C14.S1.XMD4.m1adec
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

Latest revision as of 12:42, 2 January 2020

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English
DLMF:14.25.E2
No description defined

    Statements

    test
    𝑸 ν μ ⁡ ( z ) = π 1 / 2 ⁢ ( z 2 - 1 ) μ / 2 2 μ ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( z + ( z 2 - 1 ) 1 / 2 ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t , associated-Legendre-black-Q 𝜇 𝜈 𝑧 superscript 𝜋 1 2 superscript superscript 𝑧 2 1 𝜇 2 superscript 2 𝜇 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 𝜇 1 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝑧 superscript superscript 𝑧 2 1 1 2 𝑡 𝜈 𝜇 1 𝑡 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi% ^{1/2}\left(z^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(% z+(z^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.25.E2
    0 references
    constraint
    ℜ ⁡ ( ν + 1 ) > ℜ ⁡ μ > - 1 2 𝜈 1 𝜇 1 2 {\displaystyle{\displaystyle\Re(\nu+1)>\Re\mu>-\tfrac{1}{2}}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    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.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11638
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C14.S1.XMD4.m1adec
    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
     
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