DLMF:19.12.E2 (Q6995)

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DLMF:19.12.E2
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    Statements

    DLMF defining formula
    E ⁡ ( k ) = 1 + 1 2 ⁢ ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 3 2 ) m ( 2 ) m ⁢ m ! ⁢ k ′ 2 ⁢ m + 2 ⁢ ( ln ⁡ ( 1 k ′ ) + d ⁢ ( m ) - 1 ( 2 ⁢ m + 1 ) ⁢ ( 2 ⁢ m + 2 ) ) , complete-elliptic-integral-second-kind-E 𝑘 1 1 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 3 2 𝑚 Pochhammer 2 𝑚 𝑚 superscript superscript 𝑘 ′ 2 𝑚 2 1 superscript 𝑘 ′ 𝑑 𝑚 1 2 𝑚 1 2 𝑚 2 {\displaystyle{\displaystyle E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2% \right)_{m}}m!}{k^{\prime}}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)% +d(m)-\frac{1}{(2m+1)(2m+2)}\right),}}
    0 references
    constraint
    | k ′ | < 1 superscript 𝑘 ′ 1 {\displaystyle{\displaystyle|k^{\prime}|<1}}
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    Symbols used
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
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    Legendre’s complete elliptic integral of the second kind
    DLMF defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2adec
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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    principal branch of logarithm function
    DLMF defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
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    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C19.S1.XMD2.m1adec
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    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1adec
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    complementary modulus
    DLMF defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1adec
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    function
    DLMF defining formula
    d ⁢ ( m ) 𝑑 𝑚 {\displaystyle{\displaystyle d(m)}}
    xml-id
    C19.S12.XMD1.m1adec
    0 references
     
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