DLMF:14.20.E9 (Q4930): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11595 / rank
 
Normal rank
Property / Symbols used: Q11595 / qualifier
 
Defining formula:

𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}

\FerrersP[]{\NVar{\nu}}@{\NVar{x}}=\FerrersP[0]{\nu}@{x}
Property / Symbols used: Q11595 / qualifier
 
xml-id: C14.S2.SS2.p2.m2adec
Property / Symbols used
 
Property / Symbols used: Q11630 / rank
 
Normal rank
Property / Symbols used: Q11630 / qualifier
 
Defining formula:

τ 𝜏 {\displaystyle{\displaystyle\tau}}

\tau
Property / Symbols used: Q11630 / qualifier
 
xml-id: C14.S1.XMD3.m1pdec
Property / Symbols used
 
Property / Symbols used: Q11631 / rank
 
Normal rank
Property / Symbols used: Q11631 / qualifier
 
Defining formula:

0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}

0<\theta<\pi
Property / Symbols used: Q11631 / qualifier
 
xml-id: C14.S20.XMD1.m1dec

Latest revision as of 13:34, 2 January 2020

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DLMF:14.20.E9
No description defined

    Statements

    Defining formula
    𝖯 - 1 2 + i ⁢ τ ⁡ ( cos ⁡ θ ) = 2 π ⁢ ∫ 0 θ cosh ⁡ ( τ ⁢ ϕ ) 2 ⁢ ( cos ⁡ ϕ - cos ⁡ θ ) ⁢ d ϕ . shorthand-Ferrers-Legendre-P-first-kind 1 2 𝑖 𝜏 𝜃 2 𝜋 superscript subscript 0 𝜃 𝜏 italic-ϕ 2 italic-ϕ 𝜃 italic-ϕ {\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta% \right)=\frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{% 2(\cos\phi-\cos\theta)}}\mathrm{d}\phi.}}
    0 references
    DLMF id
    DLMF:14.20.E9
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aodec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11595
    Defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2adec
    0 references
    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1pdec
    0 references
    Q11631
    Defining formula
    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
    xml-id
    C14.S20.XMD1.m1dec
    0 references
     
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