DLMF:19.5.E2 (Q6131)

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DLMF:19.5.E2
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    Statements

    Defining formula
    E ⁡ ( k ) = π 2 ⁢ ∑ m = 0 ∞ ( - 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m = π 2 ⁢ F 1 2 ⁡ ( - 1 2 , 1 2 ; 1 ; k 2 ) , complete-elliptic-integral-second-kind-E 𝑘 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 𝜋 2 Gauss-hypergeometric-F-as-2F1 1 2 1 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}% \frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}% k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}% \right),}}
    0 references
    DLMF id
    DLMF:19.5.E2
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5aadec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    Q11818
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C19.S1.XMD2.m1adec
    0 references
     
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