DLMF:10.9.E25 (Q3095)

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DLMF:10.9.E25
No description defined

    Statements

    Defining formula
    H ν ( 2 ) ⁡ ( 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-2-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^{(2)}_{\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.E25
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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
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
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    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.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.m2axdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1axdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2andec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2amdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3axdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1tdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1tdec
    0 references
    Q11436
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C10.S9.XMD1.m1bdec
    0 references
     
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