DLMF:10.22.E17 (Q3391): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the second kind / qualifier
 
Defining formula:

Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselY{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the second kind / qualifier
 
xml-id: C10.S2.E3.m2adec

Revision as of 01:11, 2 January 2020

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DLMF:10.22.E17
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π Y 2 ⁢ ν ⁡ ( 2 ⁢ z ⁢ cos ⁡ θ ) ⁢ cos ⁡ ( 2 ⁢ μ ⁢ θ ) ⁢ d θ = 1 2 ⁢ π ⁢ cot ⁡ ( 2 ⁢ ν ⁢ π ) ⁢ J ν + μ ⁡ ( z ) ⁢ J ν - μ ⁡ ( z ) - 1 2 ⁢ π ⁢ csc ⁡ ( 2 ⁢ ν ⁢ π ) ⁢ J μ - ν ⁡ ( z ) ⁢ J - μ - ν ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-Y-Weber 2 𝜈 2 𝑧 𝜃 2 𝜇 𝜃 𝜃 1 2 𝜋 2 𝜈 𝜋 Bessel-J 𝜈 𝜇 𝑧 Bessel-J 𝜈 𝜇 𝑧 1 2 𝜋 2 𝜈 𝜋 Bessel-J 𝜇 𝜈 𝑧 Bessel-J 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos% \theta\right)\cos\left(2\mu\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi\cot% \left(2\nu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{% 1}{2}\pi\csc\left(2\nu\pi\right)J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z% \right),}}
    0 references
    DLMF id
    DLMF:10.22.E17
    0 references
    constraint
    - 1 2 < ℜ ⁡ ν < 1 2 1 2 𝜈 1 2 {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}}}
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aidec
    0 references
    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
     
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