DLMF:10.17.E13 (Q3182): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2addec

Revision as of 15:31, 2 January 2020

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DLMF:10.17.E13
No description defined

    Statements

    Defining formula
    H ν ( 1 ) ⁡ ( z ) H ν ( 2 ) ⁡ ( z ) } = ( 2 π ⁢ z ) 1 2 ⁢ e ± i ⁢ ω ⁢ ( ∑ k = 0 ℓ - 1 ( ± i ) k ⁢ a k ⁢ ( ν ) z k + R ℓ ± ⁢ ( ν , z ) ) , selection Hankel-H-1-Bessel-third-kind 𝜈 𝑧 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 2 𝜋 𝑧 1 2 superscript 𝑒 plus-or-minus 𝑖 𝜔 superscript subscript 𝑘 0 ℓ 1 superscript plus-or-minus 𝑖 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 superscript subscript 𝑅 ℓ plus-or-minus 𝜈 𝑧 {\displaystyle{\displaystyle\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{% \pm i\omega}\left(\sum_{k=0}^{\ell-1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}+R_{% \ell}^{\pm}(\nu,z)\right),}}
    0 references
    DLMF id
    DLMF:10.17.E13
    0 references
    constraint
    ℓ = 1 , 2 , … ℓ 1 2 … {\displaystyle{\displaystyle\ell=1,2,\ldots}}
    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.m2abdec
    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.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
     
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