DLMF:10.17.E11 (Q3180): Difference between revisions

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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.m2agdec

Revision as of 14:31, 2 January 2020

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DLMF:10.17.E11
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    Statements

    test
    H ν ( 1 ) ′ ⁡ ( z ) ∼ i ⁢ ( 2 π ⁢ z ) 1 2 ⁢ e i ⁢ ω ⁢ ∑ k = 0 ∞ i k ⁢ b k ⁢ ( ν ) z k , asymptotic-expansion diffop Hankel-H-1-Bessel-third-kind 𝜈 1 𝑧 𝑖 superscript 2 𝜋 𝑧 1 2 superscript 𝑒 𝑖 𝜔 superscript subscript 𝑘 0 superscript 𝑖 𝑘 subscript 𝑏 𝑘 𝜈 superscript 𝑧 𝑘 {\displaystyle{\displaystyle{H^{(1)}_{\nu}}'\left(z\right)\sim i\left(\frac{2}% {\pi z}\right)^{\frac{1}{2}}e^{i\omega}\sum_{k=0}^{\infty}i^{k}\frac{b_{k}(\nu% )}{z^{k}},}}
    0 references
    DLMF id
    DLMF:10.17.E11
    0 references
    constraint
    - π + δ ≤ ph ⁡ z ≤ 2 ⁢ π - δ 𝜋 𝛿 phase 𝑧 2 𝜋 𝛿 {\displaystyle{\displaystyle-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta}}
    0 references
    Symbols used
    Bessel function of the third kind (or Hankel function)
    test
    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
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
     
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