DLMF:13.23.E9 (Q4643)

From DRMF
Revision as of 01:03, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:13.23.E9
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - 1 2 ⁢ t ⁢ t μ - 1 2 ⁢ ( ν + 1 ) ⁢ M κ , μ ⁡ ( t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = Γ ⁡ ( 1 + 2 ⁢ μ ) Γ ⁡ ( 1 2 - μ + κ + ν ) ⁢ e - 1 2 ⁢ x ⁢ x 1 2 ⁢ ( κ - μ - 3 2 ) ⁢ M 1 2 ⁢ ( κ + 3 ⁢ μ - ν + 1 2 ) , 1 2 ⁢ ( κ - μ + ν - 1 2 ) ⁡ ( x ) , superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜇 1 2 𝜈 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 𝜈 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 𝜅 𝜇 3 2 Whittaker-confluent-hypergeometric-M 1 2 𝜅 3 𝜇 𝜈 1 2 1 2 𝜅 𝜇 𝜈 1 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{% 2}(\nu+1)}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}% t=\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu% \right)}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1% }{2}(\kappa+3\mu-\nu+\frac{1}{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}% \left(x\right),}}
    0 references
    DLMF id
    DLMF:13.23.E9
    0 references
    constraint
    - 1 2 < ℜ ⁡ μ < ℜ ⁡ ( κ + 1 2 ⁢ ν ) + 3 missing 1 2 𝜇 𝜅 1 2 𝜈 3 missing {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\mu<\Re(\kappa+\tfrac{1}{2}\nu)+% \tfrac{3}{missing}\)\@add@PDF@RDFa@triples\end{document}}}
    0 references
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    - 1 2 < ℜ ⁡ μ < ℜ ⁡ ( κ + 1 2 ⁢ ν ) + 3 4 1 2 𝜇 𝜅 1 2 𝜈 3 4 {\displaystyle{\displaystyle-\tfrac{1}{2}<\Re\mu<\Re(\kappa+\tfrac{1}{2}\nu)+% \tfrac{3}{4}}}
    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.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4643&oldid=29093"