DLMF:10.9.E24 (Q3094)

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DLMF:10.9.E24
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    Statements

    Defining formula
    H ν ( 1 ) ⁡ ( z ) = - e - 1 2 ⁢ ν ⁢ π ⁢ i 2 ⁢ π 2 ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( t - ν ) ⁢ ( - 1 2 ⁢ i ⁢ z ) ν - 2 ⁢ t ⁢ d t , Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 2 superscript 𝜋 2 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 Euler-Gamma 𝑡 𝜈 superscript 1 2 𝑖 𝑧 𝜈 2 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}\left(z\right)=-\frac{e^{-\frac{1}{% 2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma% \left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E24
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    constraint
    0 < ph ⁡ z < π 0 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<\operatorname{ph}z<\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2awdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1awdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2amdec
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