DLMF:10.22.E62 (Q3436)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.22.E62
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ t μ - ν + 1 ⁢ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = { 0 , 0 < b < a , 2 μ - ν + 1 ⁢ a μ ⁢ ( b 2 - a 2 ) ν - μ - 1 b ν ⁢ Γ ⁡ ( ν - μ ) , 0 < a ≤ b . superscript subscript 0 superscript 𝑡 𝜇 𝜈 1 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 cases 0 0 𝑏 𝑎 superscript 2 𝜇 𝜈 1 superscript 𝑎 𝜇 superscript superscript 𝑏 2 superscript 𝑎 2 𝜈 𝜇 1 superscript 𝑏 𝜈 Euler-Gamma 𝜈 𝜇 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-\nu+1}J_{\mu}\left(at% \right)J_{\nu}\left(bt\right)\mathrm{d}t=\begin{cases}0,&0<b<a,\\ \dfrac{2^{\mu-\nu+1}a^{\mu}(b^{2}-a^{2})^{\nu-\mu-1}}{b^{\nu}\Gamma\left(\nu-% \mu\right)},&0<a\leq b.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E62
    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.m2aaudec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2atdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abidec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abidec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ardec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3436&oldid=36375"