DLMF:10.17.E12 (Q3181)

From DRMF
Revision as of 15:31, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.17.E12
No description defined

    Statements

    Defining formula
    H ν ( 2 ) ′ ⁡ ( z ) ∼ - i ⁢ ( 2 π ⁢ z ) 1 2 ⁢ e - i ⁢ ω ⁢ ∑ k = 0 ∞ ( - i ) k ⁢ b k ⁢ ( ν ) z k , asymptotic-expansion diffop Hankel-H-2-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^{(2)}_{\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.E12
    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)
    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.m2aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1idec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1idec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1jdec
    0 references
    Q11442
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C10.S1.XMD8.m1gdec
    0 references
    DLMF:10.17.E2
    Defining formula
    ω 𝜔 {\displaystyle{\displaystyle\omega}}
    xml-id
    C10.S17.XMD2.m1hdec
    0 references
    expansion (locally)
    Defining formula
    b k ⁢ ( ν ) subscript 𝑏 𝑘 𝜈 {\displaystyle{\displaystyle b_{k}(\nu)}}
    xml-id
    C10.S17.XMD3.m1ddec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3181&oldid=58547"