DLMF:14.20.E14 (Q4935)

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DLMF:14.20.E14
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    Statements

    Defining formula
    π ⁢ ∫ 0 ∞ τ ⁢ tanh ⁡ ( τ ⁢ π ) cosh ⁡ ( τ ⁢ π ) ⁢ P - 1 2 + i ⁢ τ ⁡ ( x ) ⁢ P - 1 2 + i ⁢ τ ⁡ ( y ) ⁢ d τ = 1 y + x . 𝜋 superscript subscript 0 𝜏 𝜏 𝜋 𝜏 𝜋 shorthand-Legendre-P-first-kind 1 2 𝑖 𝜏 𝑥 shorthand-Legendre-P-first-kind 1 2 𝑖 𝜏 𝑦 𝜏 1 𝑦 𝑥 {\displaystyle{\displaystyle\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi% \right)}{\cosh\left(\tau\pi\right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-% \frac{1}{2}+i\tau}\left(y\right)\mathrm{d}\tau=\frac{1}{y+x}.}}
    0 references
    DLMF id
    DLMF:14.20.E14
    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.m2aidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aedec
    0 references
    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
    0 references
     
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