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

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Property / constraint
 

ℓ = 1 , 2 , … ℓ 1 2 … {\displaystyle{\displaystyle\ell=1,2,\ldots}}

\ell=1,2,\dotsc
Property / constraint: ℓ = 1 , 2 , … ℓ 1 2 … {\displaystyle{\displaystyle\ell=1,2,\ldots}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Bessel function of the third kind (or Hankel function) / rank
 
Normal rank
Property / Symbols used: Bessel function of the third kind (or Hankel function) / qualifier
 
Defining formula:

H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}

\HankelH{1}{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the third kind (or Hankel function) / qualifier
 
xml-id: C10.S2.E5.m2abdec
Property / Symbols used
 
Property / Symbols used: Bessel function of the third kind (or Hankel function) / rank
 
Normal rank
Property / Symbols used: Bessel function of the third kind (or Hankel function) / qualifier
 
Defining formula:

H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}

\HankelH{2}{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the third kind (or Hankel function) / qualifier
 
xml-id: C10.S2.E6.m2abdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aidec
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
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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1kdec
Property / Symbols used
 
Property / Symbols used: expansion (locally) / rank
 
Normal rank
Property / Symbols used: expansion (locally) / qualifier
 
Defining formula:

a k ⁢ ( ν ) subscript 𝑎 𝑘 𝜈 {\displaystyle{\displaystyle a_{k}(\nu)}}

a_{k}(\nu)
Property / Symbols used: expansion (locally) / qualifier
 
xml-id: C10.S17.XMD1.m1edec
Property / Symbols used
 
Property / Symbols used: DLMF:10.17.E2 / rank
 
Normal rank
Property / Symbols used: DLMF:10.17.E2 / qualifier
 
Defining formula:

ω 𝜔 {\displaystyle{\displaystyle\omega}}

\omega
Property / Symbols used: DLMF:10.17.E2 / qualifier
 
xml-id: C10.S17.XMD2.m1idec
Property / Symbols used
 
Property / Symbols used: remainder (locally) / rank
 
Normal rank
Property / Symbols used: remainder (locally) / qualifier
 
Defining formula:

R ℓ ± ⁢ ( ν , z ) superscript subscript 𝑅 ℓ plus-or-minus 𝜈 𝑧 {\displaystyle{\displaystyle R_{\ell}^{\pm}(\nu,z)}}

R_{\ell}^{\pm}(\nu,z)
Property / Symbols used: remainder (locally) / qualifier
 
xml-id: C10.S17.XMD4.m1dec

Latest revision as of 15:32, 2 January 2020

No description defined
Language Label Description Also known as
English
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
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1jdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1jdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1kdec
    0 references
    expansion (locally)
    Defining formula
    a k ⁢ ( ν ) subscript 𝑎 𝑘 𝜈 {\displaystyle{\displaystyle a_{k}(\nu)}}
    xml-id
    C10.S17.XMD1.m1edec
    0 references
    DLMF:10.17.E2
    Defining formula
    ω 𝜔 {\displaystyle{\displaystyle\omega}}
    xml-id
    C10.S17.XMD2.m1idec
    0 references
    remainder (locally)
    Defining formula
    R ℓ ± ⁢ ( ν , z ) superscript subscript 𝑅 ℓ plus-or-minus 𝜈 𝑧 {\displaystyle{\displaystyle R_{\ell}^{\pm}(\nu,z)}}
    xml-id
    C10.S17.XMD4.m1dec
    0 references
     
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