DLMF:10.17.E6 (Q3175): Difference between revisions

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Property / constraint
 

- 2 ⁢ π + δ ≤ ph ⁡ z ≤ π - δ 2 𝜋 𝛿 phase 𝑧 𝜋 𝛿 {\displaystyle{\displaystyle-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta}}

-2\pi+\delta\leq\phase@@{z}\leq\pi-\delta
Property / constraint: - 2 ⁢ π + δ ≤ ph ⁡ z ≤ π - δ 2 𝜋 𝛿 phase 𝑧 𝜋 𝛿 {\displaystyle{\displaystyle-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta}} / 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
 
test:

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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
test:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11acdec
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
 
test:

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.m2addec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1edec
Property / Symbols used
 
Property / Symbols used: Q11442 / rank
 
Normal rank
Property / Symbols used: Q11442 / qualifier
 
test:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11442 / qualifier
 
xml-id: C10.S1.XMD8.m1cdec
Property / Symbols used
 
Property / Symbols used: expansion (locally) / rank
 
Normal rank
Property / Symbols used: expansion (locally) / qualifier
 
test:

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.m1ddec
Property / Symbols used
 
Property / Symbols used: DLMF:10.17.E2 / rank
 
Normal rank
Property / Symbols used: DLMF:10.17.E2 / qualifier
 
test:

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

\omega
Property / Symbols used: DLMF:10.17.E2 / qualifier
 
xml-id: C10.S17.XMD2.m1ddec

Latest revision as of 14:29, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.17.E6
No description defined

    Statements

    test
    H ν ( 2 ) ⁡ ( z ) ∼ ( 2 π ⁢ z ) 1 2 ⁢ e - i ⁢ ω ⁢ ∑ k = 0 ∞ ( - i ) k ⁢ a k ⁢ ( ν ) z k , asymptotic-expansion Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 2 𝜋 𝑧 1 2 superscript 𝑒 𝑖 𝜔 superscript subscript 𝑘 0 superscript 𝑖 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 {\displaystyle{\displaystyle{H^{(2)}_{\nu}}\left(z\right)\sim\left(\frac{2}{% \pi z}\right)^{\frac{1}{2}}e^{-i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{a_{k}(% \nu)}{z^{k}},}}
    0 references
    DLMF id
    DLMF:10.17.E6
    0 references
    constraint
    - 2 ⁢ π + δ ≤ ph ⁡ z ≤ π - δ 2 𝜋 𝛿 phase 𝑧 𝜋 𝛿 {\displaystyle{\displaystyle-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta}}
    0 references
    Symbols used
    Bessel function of the third kind (or Hankel function)
    test
    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
    0 references
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1ddec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ddec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1edec
    0 references
    Q11442
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C10.S1.XMD8.m1cdec
    0 references
    expansion (locally)
    test
    a k ⁢ ( ν ) subscript 𝑎 𝑘 𝜈 {\displaystyle{\displaystyle a_{k}(\nu)}}
    xml-id
    C10.S17.XMD1.m1ddec
    0 references
    DLMF:10.17.E2
    test
    ω 𝜔 {\displaystyle{\displaystyle\omega}}
    xml-id
    C10.S17.XMD2.m1ddec
    0 references
     
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