DLMF:8.6.E6 (Q2520)

From DRMF
Revision as of 13:43, 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:8.6.E6
No description defined

    Statements

    Defining formula
    Γ ⁡ ( a , z ) = 2 ⁢ z 1 2 ⁢ a ⁢ e - z Γ ⁡ ( 1 - a ) ⁢ ∫ 0 ∞ e - t ⁢ t - 1 2 ⁢ a ⁢ K a ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t , incomplete-Gamma 𝑎 𝑧 2 superscript 𝑧 1 2 𝑎 superscript 𝑒 𝑧 Euler-Gamma 1 𝑎 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 1 2 𝑎 modified-Bessel-second-kind 𝑎 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{2z^{\frac{1}{2}a}e^{-% z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(% 2\sqrt{zt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:8.6.E6
    0 references
    constraint
    ℜ ⁡ a < 1 𝑎 1 {\displaystyle{\displaystyle\Re a<1}}
    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.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    incomplete gamma function
    Defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1edec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2520&oldid=39350"