DLMF:14.15.E12 (Q4858): 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
 
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.m2addec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11091 / rank
 
Normal rank
Property / Symbols used: Q11091 / qualifier
 
Defining formula:

env ⁢ Y ν ⁡ ( x ) envelope-Bessel-Y 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar% {x}\right)}}

\envBesselY{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Q11091 / qualifier
 
xml-id: C2.S8.SS4.p5.m4adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11615 / rank
 
Normal rank
Property / Symbols used: Q11615 / qualifier
 
Defining formula:

θ 𝜃 {\displaystyle{\displaystyle\theta}}

\theta
Property / Symbols used: Q11615 / qualifier
 
xml-id: C14.S15.XMD7.m1adec

Latest revision as of 01:49, 2 January 2020

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DLMF:14.15.E12
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    Statements

    Defining formula
    𝖰 ν - μ ⁡ ( cos ⁡ θ ) = - π 2 ⁢ ν μ ⁢ ( θ sin ⁡ θ ) 1 / 2 ⁢ ( Y μ ⁡ ( ( ν + 1 2 ) ⁢ θ ) + O ⁡ ( 1 ν ) ⁢ env ⁢ Y μ ⁡ ( ( ν + 1 2 ) ⁢ θ ) ) , Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝜃 𝜋 2 superscript 𝜈 𝜇 superscript 𝜃 𝜃 1 2 Bessel-Y-Weber 𝜇 𝜈 1 2 𝜃 Big-O 1 𝜈 envelope-Bessel-Y 𝜇 𝜈 1 2 𝜃 {\displaystyle{\displaystyle\mathsf{Q}^{-\mu}_{\nu}\left(\cos\theta\right)=-% \frac{\pi}{2\nu^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}\left(Y_{\mu% }\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)+O\left(\frac{1}{\nu}\right)% \mathrm{env}\mskip-2.0mu Y_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)\theta% \right)\right),}}
    0 references
    DLMF id
    DLMF:14.15.E12
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2adec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2agdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.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.m2addec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q11091
    Defining formula
    env ⁢ Y ν ⁡ ( x ) envelope-Bessel-Y 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu Y_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m4adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1hdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1hdec
    0 references
    Q11615
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C14.S15.XMD7.m1adec
    0 references
     
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