DLMF:11.10.E3 (Q4020): Difference between revisions

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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.m3abdec

Revision as of 14:54, 2 January 2020

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DLMF:11.10.E3
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    Statements

    Defining formula
    1 π ⁢ ∫ 0 2 ⁢ π cos ⁡ ( ν ⁢ θ - z ⁢ sin ⁡ θ ) ⁢ d θ = ( 1 + cos ⁡ ( 2 ⁢ π ⁢ ν ) ) ⁢ 𝐉 ν ⁡ ( z ) + sin ⁡ ( 2 ⁢ π ⁢ ν ) ⁢ 𝐄 ν ⁡ ( z ) . 1 𝜋 superscript subscript 0 2 𝜋 𝜈 𝜃 𝑧 𝜃 𝜃 1 2 𝜋 𝜈 Anger-J 𝜈 𝑧 2 𝜋 𝜈 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{2\pi}\cos\left(\nu\theta-z% \sin\theta\right)\mathrm{d}\theta=(1+\cos\left(2\pi\nu\right))\,\mathbf{J}_{% \nu}\left(z\right)+\sin\left(2\pi\nu\right)\mathbf{E}_{\nu}\left(z\right).}}
    0 references
    DLMF id
    DLMF:11.10.E3
    0 references
    Symbols used
    Anger function
    Defining formula
    𝐉 ν ⁡ ( z ) Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S10.E1.m2aadec
    0 references
    Weber function
    Defining formula
    𝐄 ν ⁡ ( z ) Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S10.E2.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
     
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