DLMF:13.11.E1 (Q4476): 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: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec

Revision as of 16:02, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:13.11.E1
No description defined

    Statements

    Defining formula
    M ⁡ ( a , b , z ) = Γ ⁡ ( a - 1 2 ) ⁢ e 1 2 ⁢ z ⁢ ( 1 4 ⁢ z ) 1 2 - a ⁢ ∑ s = 0 ∞ ( 2 ⁢ a - 1 ) s ⁢ ( 2 ⁢ a - b ) s ( b ) s ⁢ s ! ⁢ ( a - 1 2 + s ) ⁢ I a - 1 2 + s ⁡ ( 1 2 ⁢ z ) , Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑎 1 2 superscript 𝑒 1 2 𝑧 superscript 1 4 𝑧 1 2 𝑎 superscript subscript 𝑠 0 Pochhammer 2 𝑎 1 𝑠 Pochhammer 2 𝑎 𝑏 𝑠 Pochhammer 𝑏 𝑠 𝑠 𝑎 1 2 𝑠 modified-Bessel-first-kind 𝑎 1 2 𝑠 1 2 𝑧 {\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}% \right)e^{\frac{1}{2}z}\left(\tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^% {\infty}\frac{{\left(2a-1\right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_% {s}}s!}\*\left(a-\tfrac{1}{2}+s\right)\*I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z% \right),}}
    0 references
    DLMF id
    DLMF:13.11.E1
    0 references
    constraint
    a + 1 2 , b ≠ 0 , - 1 , - 2 , … formulae-sequence 𝑎 1 2 𝑏 0 1 2 … {\displaystyle{\displaystyle a+\frac{1}{2},b\neq 0,-1,-2,\dots}}
    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.m2adec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4476&oldid=62371"