DLMF:10.9.E16 (Q3084)

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DLMF:10.9.E16
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    Statements

    Defining formula
    ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ H ν ( 2 ) ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = - 1 π ⁢ i ⁢ e 1 2 ⁢ ν ⁢ π ⁢ i ⁢ ∫ - ∞ ∞ e - i ⁢ z ⁢ cosh ⁡ t - i ⁢ ζ ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t , superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Hankel-H-2-Bessel-third-kind 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 𝑖 superscript 𝑒 1 2 𝜈 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝑖 𝜁 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}{H^{(2)}_{\nu}}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i% }e^{\frac{1}{2}\nu\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t% }\mathrm{d}t,}}
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    DLMF id
    DLMF:10.9.E16
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    constraint
    ℑ ⁡ ( z ± ζ ) < 0 plus-or-minus 𝑧 𝜁 0 {\displaystyle{\displaystyle\Im(z\pm\zeta)<0}}
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    Symbols used
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.m2aadec
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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.m2aodec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
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