DLMF:10.22.E26 (Q3400): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2aldec

Revision as of 01:18, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E26
No description defined

    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin ⁡ θ ) ⁢ J ν ⁡ ( ζ ⁢ cos ⁡ θ ) ⁢ ( sin ⁡ θ ) μ + 1 ⁢ ( cos ⁡ θ ) ν + 1 ⁢ d θ = z μ ⁢ ζ ν ⁢ J μ + ν + 1 ⁡ ( ζ 2 + z 2 ) ( ζ 2 + z 2 ) 1 2 ⁢ ( μ + ν + 1 ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 𝜃 Bessel-J 𝜈 𝜁 𝜃 superscript 𝜃 𝜇 1 superscript 𝜃 𝜈 1 𝜃 superscript 𝑧 𝜇 superscript 𝜁 𝜈 Bessel-J 𝜇 𝜈 1 superscript 𝜁 2 superscript 𝑧 2 superscript superscript 𝜁 2 superscript 𝑧 2 1 2 𝜇 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)J_{\nu}\left(\zeta\cos\theta\right)(\sin\theta)^{\mu+1}(\cos\theta)^{% \nu+1}\mathrm{d}\theta=\frac{z^{\mu}\zeta^{\nu}J_{\mu+\nu+1}\left(\sqrt{\zeta^% {2}+z^{2}}\right)}{(\zeta^{2}+z^{2})^{\frac{1}{2}(\mu+\nu+1)}},}}
    0 references
    DLMF id
    DLMF:10.22.E26
    0 references
    constraint
    ℜ ⁡ μ > - 1 , ℜ ⁡ ν > - 1 formulae-sequence 𝜇 1 𝜈 1 {\displaystyle{\displaystyle\Re\mu>-1,\Re\nu>-1}}
    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.m2ardec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aldec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3400&oldid=30270"