DLMF:10.61.E2 (Q3791)

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DLMF:10.61.E2
No description defined

    Statements

    Defining formula
    ker ν ⁡ x + i ⁢ kei ν ⁡ x = e - ν ⁢ π ⁢ i / 2 ⁢ K ν ⁡ ( x ⁢ e π ⁢ i / 4 ) = 1 2 ⁢ π ⁢ i ⁢ H ν ( 1 ) ⁡ ( x ⁢ e 3 ⁢ π ⁢ i / 4 ) = - 1 2 ⁢ π ⁢ i ⁢ e - ν ⁢ π ⁢ i ⁢ H ν ( 2 ) ⁡ ( x ⁢ e - π ⁢ i / 4 ) . Kelvin-ker 𝜈 𝑥 𝑖 Kelvin-kei 𝜈 𝑥 superscript 𝑒 𝜈 𝜋 𝑖 2 modified-Bessel-second-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 1 2 𝜋 𝑖 Hankel-H-1-Bessel-third-kind 𝜈 𝑥 superscript 𝑒 3 𝜋 𝑖 4 1 2 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 Hankel-H-2-Bessel-third-kind 𝜈 𝑥 superscript 𝑒 𝜋 𝑖 4 {\displaystyle{\displaystyle\operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu% }x=e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)=\tfrac{1}{2}\pi i{H^{(1)}_{% \nu}}\left(xe^{3\pi i/4}\right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}{H^{(2)}_{\nu}}% \left(xe^{-\pi i/4}\right).}}
    0 references
    DLMF id
    DLMF:10.61.E2
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    Symbols defined
    Kelvin function
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    Kelvin function
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    Symbols used
    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.m2adec
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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.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.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
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    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1adec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1adec
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