DLMF:10.41.E12 (Q3609): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m1fdec

Revision as of 13:57, 2 January 2020

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DLMF:10.41.E12
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    Statements

    Defining formula
    I ν ⁡ ( ν ⁢ z ) = e ν ⁢ η ( 2 ⁢ π ⁢ ν ) 1 2 ⁢ ( 1 + z 2 ) 1 4 ⁢ ( ∑ k = 0 ℓ - 1 U k ⁢ ( p ) ν k + O ⁡ ( 1 z ℓ ) ) , modified-Bessel-first-kind 𝜈 𝜈 𝑧 superscript 𝑒 𝜈 𝜂 superscript 2 𝜋 𝜈 1 2 superscript 1 superscript 𝑧 2 1 4 superscript subscript 𝑘 0 ℓ 1 subscript 𝑈 𝑘 𝑝 superscript 𝜈 𝑘 Big-O 1 superscript 𝑧 ℓ {\displaystyle{\displaystyle I_{\nu}\left(\nu z\right)=\frac{e^{\nu\eta}}{(2% \pi\nu)^{\frac{1}{2}}(1+z^{2})^{\frac{1}{4}}}\left(\sum_{k=0}^{\ell-1}\frac{U_% {k}(p)}{\nu^{k}}+O\left(\frac{1}{z^{\ell}}\right)\right),}}
    0 references
    DLMF id
    DLMF:10.41.E12
    0 references
    constraint
    | ph ⁡ z | ≤ 1 2 ⁢ π - δ phase 𝑧 1 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta}}
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2acdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1fdec
    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.m1fdec
    0 references
     
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