DLMF:19.4.E3 (Q6123)

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DLMF:19.4.E3
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    Statements

    Defining formula
    d 2 E ⁡ ( k ) d k 2 = - 1 k ⁢ d K ⁡ ( k ) d k = k ′ 2 ⁢ K ⁡ ( k ) - E ⁡ ( k ) k 2 ⁢ k ′ 2 , derivative complete-elliptic-integral-second-kind-E 𝑘 𝑘 2 1 𝑘 derivative complete-elliptic-integral-first-kind-K 𝑘 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}% k}^{2}}=-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{% \prime}}^{2}K\left(k\right)-E\left(k\right)}{k^{2}{k^{\prime}}^{2}},}}
    0 references
    DLMF id
    DLMF:19.4.E3
    0 references
    Symbols used
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1abdec
    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.m2abdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2abdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1bdec
    0 references
     
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