DLMF:10.4.E8 (Q3023): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2addec

Revision as of 15:06, 2 January 2020

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DLMF:10.4.E8
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    Statements

    Defining formula
    H ν ( 2 ) ⁡ ( z ) = i ⁢ csc ⁡ ( ν ⁢ π ) ⁢ ( J - ν ⁡ ( z ) - e ν ⁢ π ⁢ i ⁢ J ν ⁡ ( z ) ) = csc ⁡ ( ν ⁢ π ) ⁢ ( Y - ν ⁡ ( z ) - e ν ⁢ π ⁢ i ⁢ Y ν ⁡ ( z ) ) . Hankel-H-2-Bessel-third-kind 𝜈 𝑧 𝑖 𝜈 𝜋 Bessel-J 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-J 𝜈 𝑧 𝜈 𝜋 Bessel-Y-Weber 𝜈 𝑧 superscript 𝑒 𝜈 𝜋 𝑖 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi% \right)\left(J_{-\nu}\left(z\right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=% \csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z% \right)\right).}}
    0 references
    DLMF id
    DLMF:10.4.E8
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aedec
    0 references
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aedec
    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.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.m2acdec
    0 references
    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2abdec
    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.m2addec
    0 references
     
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