DLMF:14.20.E9 (Q4930)

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DLMF:14.20.E9
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    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
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2acdec
    0 references
     
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