DLMF:10.22.E13 (Q3387): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aldec

Revision as of 15:50, 2 January 2020

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DLMF:10.22.E13
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J 2 ⁢ ν ⁡ ( 2 ⁢ z ⁢ cos ⁡ θ ) ⁢ cos ⁡ ( 2 ⁢ μ ⁢ θ ) ⁢ d θ = 1 2 ⁢ π ⁢ J ν + μ ⁡ ( z ) ⁢ J ν - μ ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 2 𝜈 2 𝑧 𝜃 2 𝜇 𝜃 𝜃 1 2 𝜋 Bessel-J 𝜈 𝜇 𝑧 Bessel-J 𝜈 𝜇 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos% \theta\right)\cos\left(2\mu\theta\right)\mathrm{d}\theta=\tfrac{1}{2}\pi J_{% \nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right),}}
    0 references
    DLMF id
    DLMF:10.22.E13
    0 references
    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\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.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
    0 references
     
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