DLMF:10.27.E8 (Q3498): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
Defining formula:

K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2addec

Revision as of 13:36, 2 January 2020

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DLMF:10.27.E8
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    Statements

    Defining formula
    K ν ⁡ ( z ) = { 1 2 ⁢ π ⁢ i ⁢ e ν ⁢ π ⁢ i / 2 ⁢ H ν ( 1 ) ⁡ ( z ⁢ e π ⁢ i / 2 ) , - π ≤ ph ⁡ z ≤ 1 2 ⁢ π , - 1 2 ⁢ π ⁢ i ⁢ e - ν ⁢ π ⁢ i / 2 ⁢ H ν ( 2 ) ⁡ ( z ⁢ e - π ⁢ i / 2 ) , - 1 2 ⁢ π ≤ ph ⁡ z ≤ π . modified-Bessel-second-kind 𝜈 𝑧 cases 1 2 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 2 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 𝜋 phase 𝑧 1 2 𝜋 1 2 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 2 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝜋 𝑖 2 1 2 𝜋 phase 𝑧 𝜋 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\begin{cases}\tfrac{1}{2}% \pi ie^{\nu\pi i/2}{H^{(1)}_{\nu}}\left(ze^{\pi i/2}\right),&-\pi\leq% \operatorname{ph}z\leq\tfrac{1}{2}\pi,\\ -\tfrac{1}{2}\pi ie^{-\nu\pi i/2}{H^{(2)}_{\nu}}\left(ze^{-\pi i/2}\right),&-% \tfrac{1}{2}\pi\leq\operatorname{ph}z\leq\pi.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.27.E8
    0 references
    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.m2aadec
    0 references
    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
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    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.m2addec
    0 references
     
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