DLMF:13.10.E9 (Q4468)

From DRMF
Revision as of 16:01, 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:13.10.E9
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e t ⁢ z ⁢ t - a ⁢ U ⁡ ( a , b , y / t ) ⁢ d t = 2 ⁢ z 1 2 ⁢ ( 2 ⁢ a - b - 1 ) ⁢ y 1 2 ⁢ ( 1 - b ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) ⁢ K b - 1 ⁡ ( 2 ⁢ z ⁢ y ) , 1 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑡 𝑧 superscript 𝑡 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑦 𝑡 𝑡 2 superscript 𝑧 1 2 2 𝑎 𝑏 1 superscript 𝑦 1 2 1 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑦 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)% }y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}% \left(2\sqrt{zy}\right),}}
    0 references
    DLMF id
    DLMF:13.10.E9
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2acdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4468&oldid=62250"