DLMF:19.4.E3 (Q6123): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Q11601 / rank
 
Normal rank
Property / Symbols used: Q11601 / qualifier
 
Defining formula:

E ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}

\compellintEk@{\NVar{k}}
Property / Symbols used: Q11601 / qualifier
 
xml-id: C19.S2.E8.m2abdec
Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11815 / rank
 
Normal rank
Property / Symbols used: Q11815 / qualifier
 
Defining formula:

k superscript 𝑘 {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11815 / qualifier
 
xml-id: C19.S1.XMD6.m1bdec

Latest revision as of 13:37, 2 January 2020

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DLMF:19.4.E3
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    Statements

    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:19.4.E3
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    K ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    C19.S2.E8.m1abdec
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    E ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    C19.S2.E8.m2abdec
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    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    C1.S4.E4.m2abdec
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    k 𝑘 {\displaystyle{\displaystyle k}}
    C19.S1.XMD5.m1bdec
    0 references
    k superscript 𝑘 {\displaystyle{\displaystyle k^{\prime}}}
    C19.S1.XMD6.m1bdec
    0 references