DLMF:14.12.E3 (Q4824): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

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

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2adec

Revision as of 01:42, 2 January 2020

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DLMF:14.12.E3
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    Statements

    Defining formula
    𝖰 ν μ ⁡ ( cos ⁡ θ ) = π 1 / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( sin ⁡ θ ) μ 2 μ + 1 ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ( ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( cos ⁡ θ + i ⁢ sin ⁡ θ ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t + ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( cos ⁡ θ - i ⁢ sin ⁡ θ ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t ) , Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝜃 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 superscript 𝜃 𝜇 superscript 2 𝜇 1 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 𝜇 1 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝜃 𝑖 𝜃 𝑡 𝜈 𝜇 1 𝑡 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝜃 𝑖 𝜃 𝑡 𝜈 𝜇 1 𝑡 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma% \left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{% \infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}% \mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta% \cosh t)^{\nu+\mu+1}}\mathrm{d}t\right),}}
    0 references
    DLMF id
    DLMF:14.12.E3
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    constraint
    ℜ ⁡ μ > - 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
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    ℜ ⁡ ( ν ± μ ) > - 1 plus-or-minus 𝜈 𝜇 1 {\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}
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    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
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    ℜ ⁡ μ > - 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
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    ℜ ⁡ ( ν ± μ ) > - 1 plus-or-minus 𝜈 𝜇 1 {\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
     
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