DLMF:10.9.E11 (Q3078)

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DLMF:10.9.E11
No description defined

    Statements

    Defining formula
    H ν ( 2 ) ⁡ ( z ) = - e 1 2 ⁢ ν ⁢ π ⁢ i π ⁢ i ⁢ ∫ - ∞ ∞ e - i ⁢ z ⁢ cosh ⁡ t - ν ⁢ t ⁢ d t , Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 1 2 𝜈 𝜋 𝑖 𝜋 𝑖 superscript subscript superscript 𝑒 𝑖 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2% }\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{-iz\cosh t-\nu t}\mathrm{d}t,}}
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    DLMF id
    DLMF:10.9.E11
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    constraint
    - π < ph ⁡ z < 0 𝜋 phase 𝑧 0 {\displaystyle{\displaystyle-\pi<\operatorname{ph}z<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.m2adec
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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.m2ajdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2acdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1hdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1fdec
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