DLMF:10.22.E19 (Q3393): Difference between revisions

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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.m1kdec

Latest revision as of 01:12, 2 January 2020

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DLMF:10.22.E19
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin ⁡ θ ) ⁢ ( sin ⁡ θ ) μ + 1 ⁢ ( cos ⁡ θ ) 2 ⁢ ν + 1 ⁢ d θ = 2 ν ⁢ Γ ⁡ ( ν + 1 ) ⁢ z - ν - 1 ⁢ J μ + ν + 1 ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 𝜃 superscript 𝜃 𝜇 1 superscript 𝜃 2 𝜈 1 𝜃 superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript 𝑧 𝜈 1 Bessel-J 𝜇 𝜈 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)(\sin\theta)^{\mu+1}(\cos\theta)^{2\nu+1}\mathrm{d}\theta=2^{\nu}\Gamma% \left(\nu+1\right)z^{-\nu-1}J_{\mu+\nu+1}\left(z\right),}}
    0 references
    DLMF id
    DLMF:10.22.E19
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    constraint
    ℜ ⁡ μ > - 1 𝜇 1 {\displaystyle{\displaystyle\Re\mu>-1}}
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    ℜ ⁡ ν > - 1 𝜈 1 {\displaystyle{\displaystyle\Re\nu>-1}}
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    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.m2akdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ardec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1mdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1kdec
    0 references
     
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