DLMF:19.12.E1 (Q6994)

From MaRDI portal
No description defined
Language Label Description Also known as
English
DLMF:19.12.E1
No description defined

    Statements

    DLMF defining formula
    K ⁡ ( k ) = ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k ′ 2 ⁢ m ⁢ ( ln ⁡ ( 1 k ′ ) + d ⁢ ( m ) ) , complete-elliptic-integral-first-kind-K 𝑘 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript superscript 𝑘 ′ 2 𝑚 1 superscript 𝑘 ′ 𝑑 𝑚 {\displaystyle{\displaystyle K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{% 2m}\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)\right),}}
    0 references
    constraint
    0 < | k ′ | < 1 0 superscript 𝑘 ′ 1 {\displaystyle{\displaystyle 0<|k^{\prime}|<1}}
    0 references
    Symbols used
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Legendre’s complete elliptic integral of the first kind
    DLMF defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    principal branch of logarithm function
    DLMF defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C19.S1.XMD2.m1dec
    0 references
    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1dec
    0 references
    complementary modulus
    DLMF defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1dec
    0 references
    function
    DLMF defining formula
    d ⁢ ( m ) 𝑑 𝑚 {\displaystyle{\displaystyle d(m)}}
    xml-id
    C19.S12.XMD1.m1dec
    0 references

    Identifiers

     
    Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Item:Q6994&oldid=24175742"
    Powered by MediaWiki