DLMF:10.9.E13 (Q3081)

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DLMF:10.9.E13
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    Statements

    Defining formula
    ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ J ν ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ ∫ 0 π e ζ ⁢ cos ⁡ θ ⁢ cos ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - ζ ⁢ cosh ⁡ t - z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t , superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Bessel-J 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝜁 𝜃 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝜁 𝑡 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t% -\nu t}\mathrm{d}t,}}
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    DLMF id
    DLMF:10.9.E13
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    constraint
    ℜ ⁡ ( z + ζ ) > 0 𝑧 𝜁 0 {\displaystyle{\displaystyle\Re(z+\zeta)>0}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2agdec
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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.m2aldec
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