DLMF:14.19.E5 (Q4918): Difference between revisions

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imported>Admin
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Property / constraint
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m < n + 1 2 π‘š 𝑛 1 2 {\displaystyle{\displaystyle m<n+\tfrac{1}{2}}}

m<n+\tfrac{1}{2}
Property / constraint: m < n + 1 2 π‘š 𝑛 1 2 {\displaystyle{\displaystyle m<n+\tfrac{1}{2}}} / rank
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Normal rank
Property / Symbols used
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Property / Symbols used: gamma function / rank
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Normal rank
Property / Symbols used: gamma function / qualifier
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test:

Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
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xml-id: C5.S2.E1.m2abdec
Property / Symbols used
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Property / Symbols used: Q11592 / rank
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Normal rank
Property / Symbols used: Q11592 / qualifier
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test:

𝑸 Ξ½ ΞΌ ⁑ ( z ) associated-Legendre-black-Q πœ‡ 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
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xml-id: C14.S21.SS1.p1.m3aadec
Property / Symbols used
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Property / Symbols used: Q10770 / rank
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Normal rank
Property / Symbols used: Q10770 / qualifier
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test:

d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

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

cosh ⁑ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
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xml-id: C4.S28.E2.m2addec
Property / Symbols used
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Property / Symbols used: hyperbolic sine function / rank
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Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
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test:

sinh ⁑ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
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xml-id: C4.S28.E1.m2abdec
Property / Symbols used
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Property / Symbols used: Q10771 / rank
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Normal rank
Property / Symbols used: Q10771 / qualifier
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test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
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xml-id: C1.S4.SS4.m3aadec
Property / Symbols used
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Property / Symbols used: Q11593 / rank
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Normal rank
Property / Symbols used: Q11593 / qualifier
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test:

m π‘š {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11593 / qualifier
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xml-id: C14.S1.XMD5.m1adec
Property / Symbols used
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Property / Symbols used: Q11602 / rank
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Normal rank
Property / Symbols used: Q11602 / qualifier
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test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
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xml-id: C14.S1.XMD6.m1adec
Property / Symbols used
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Property / Symbols used: Q11627 / rank
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Normal rank
Property / Symbols used: Q11627 / qualifier
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test:

ΞΎ > 0 πœ‰ 0 {\displaystyle{\displaystyle\xi>0}}

\xi>0
Property / Symbols used: Q11627 / qualifier
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xml-id: C14.S19.XMD8.m1adec

Latest revision as of 12:30, 2 January 2020

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DLMF:14.19.E5
No description defined

    Statements

    test
    𝑸 n - 1 2 m ⁑ ( cosh ⁑ ΞΎ ) = Ξ“ ⁑ ( n + 1 2 ) Ξ“ ⁑ ( n + m + 1 2 ) ⁒ Ξ“ ⁑ ( n - m + 1 2 ) ⁒ ∫ 0 ∞ cosh ⁑ ( m ⁒ t ) ( cosh ⁑ ΞΎ + cosh ⁑ t ⁒ sinh ⁑ ΞΎ ) n + ( 1 / 2 ) ⁒ d t , associated-Legendre-black-Q π‘š 𝑛 1 2 πœ‰ Euler-Gamma 𝑛 1 2 Euler-Gamma 𝑛 π‘š 1 2 Euler-Gamma 𝑛 π‘š 1 2 superscript subscript 0 π‘š 𝑑 superscript πœ‰ 𝑑 πœ‰ 𝑛 1 2 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi% \right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}% \right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(% mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.19.E5
    0 references
    constraint
    m < n + 1 2 π‘š 𝑛 1 2 {\displaystyle{\displaystyle m<n+\tfrac{1}{2}}}
    0 references
    Symbols used
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q11592
    test
    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.m3aadec
    0 references
    Q10770
    test
    d x π‘₯ {\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.m2addec
    0 references
    hyperbolic sine function
    test
    sinh ⁑ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2abdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q11593
    test
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1adec
    0 references
    Q11602
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1adec
    0 references
    Q11627
    test
    ΞΎ > 0 πœ‰ 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD8.m1adec
    0 references
     
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