DLMF:10.22.E23 (Q3397): Difference between revisions

From testwiki
Jump to navigation Jump to search
← Older edit
imported>Admin
imported>Admin
 
Property / Symbols used
 
Property / Symbols used: Q11469 / rank
Normal rank
 
Property / Symbols used: Q11469 / qualifier
test:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
 
Property / Symbols used: Q11469 / qualifier
xml-id: C10.S22.XMD1.m1dec
 

Latest revision as of 01:17, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E23
No description defined

    Statements

    test
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin 2 ⁡ θ ) ⁢ J ν ⁡ ( z ⁢ cos 2 ⁡ θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ α - 1 ⁢ sec ⁡ θ ⁢ d θ = ( μ + ν + α ) ⁢ Γ ⁡ ( μ + α ) ⁢ 2 α - 1 ν ⁢ Γ ⁡ ( μ + 1 ) ⁢ z α ⁢ J μ + ν + α ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 2 𝜃 Bessel-J 𝜈 𝑧 2 𝜃 superscript 𝜃 2 𝛼 1 𝜃 𝜃 𝜇 𝜈 𝛼 Euler-Gamma 𝜇 𝛼 superscript 2 𝛼 1 𝜈 Euler-Gamma 𝜇 1 superscript 𝑧 𝛼 Bessel-J 𝜇 𝜈 𝛼 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}% \theta\right)J_{\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\alpha-1}\sec% \theta\mathrm{d}\theta=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{% \alpha-1}}{\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z% \right),}}
    0 references
    DLMF id
    DLMF:10.22.E23
    0 references
    constraint
    ℜ ⁡ ( μ + α ) > 0 𝜇 𝛼 0 {\displaystyle{\displaystyle\Re(\mu+\alpha)>0}}
    0 references
    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aodec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aidec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1avdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3avdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1ajdec
    0 references
    secant function
    test
    sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}
    xml-id
    C4.S14.E6.m2adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2ahdec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1qdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1mdec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q3397&oldid=17750"