Elliptic - Maple Help

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Elliptic Integrals

Description

 • Elliptic integrals are integrals of the form

${\int }_{a}^{b}R\left(x,\sqrt{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$

 • with R a rational function and y a polynomial of degree 3 or 4. This is the algebraic form of an elliptic integral. There are also trig forms (rational functions of sin and cos and a square root of a quadratic polynomial in sin and cos) and hyperbolic trig forms.
 • Elliptic integrals are reduced to their Legendre normal form in terms of elementary functions and the Elliptic functions EllipticF, EllipticE, and EllipticPi (or their complete versions).

Examples

Elementary answer

 > $\mathrm{int}\left(\frac{\mathrm{sqrt}\left(1+{x}^{4}\right)}{1-{x}^{4}},x=0..\frac{1}{2}\right)$
 $\frac{\sqrt{{2}}{}{\mathrm{ln}}{}\left(\sqrt{{17}}{+}{2}{}\sqrt{{2}}\right)}{{8}}{-}\frac{\sqrt{{2}}{}{\mathrm{ln}}{}\left(\sqrt{{17}}{-}{2}{}\sqrt{{2}}\right)}{{8}}{-}\frac{\sqrt{{2}}{}{\mathrm{arctan}}{}\left(\frac{\sqrt{{17}}{}\sqrt{{2}}}{{4}}\right)}{{4}}{+}\frac{{\mathrm{\pi }}{}\sqrt{{2}}}{{8}}$ (1)

Symbolic parameters

 > $\mathrm{assume}\left(0
 > $\mathrm{int}\left(\frac{{x}^{2}}{\mathrm{sqrt}\left(\left(1-{x}^{2}\right)\left(1-{k}^{2}{x}^{2}\right)\right)},x=0..k\right)$
 $\frac{{\mathrm{EllipticF}}{}\left({\mathrm{k~}}{,}{\mathrm{k~}}\right)}{{{\mathrm{k~}}}^{{2}}}{-}\frac{{\mathrm{EllipticE}}{}\left({\mathrm{k~}}{,}{\mathrm{k~}}\right)}{{{\mathrm{k~}}}^{{2}}}$ (2)

Answer as sum of roots

 > $\mathrm{ans}≔\mathrm{int}\left(\frac{1}{\left({x}^{4}+2\right)\mathrm{sqrt}\left(4-5{x}^{2}+{x}^{4}\right)},x=0..\frac{1}{4}\right)$
 ${\mathrm{ans}}{≔}\frac{\left({\sum }_{{\mathrm{_α}}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{4}}{+}{2}\right)}{}{\mathrm{EllipticPi}}{}\left(\frac{{1}}{{4}}{,}{-}\frac{{{\mathrm{_α}}}^{{2}}}{{2}}{,}\frac{{1}}{{2}}\right)\right)}{{16}}$ (3)

Can evaluate to floating point:

 > $\mathrm{evalf}\left(\mathrm{ans}\right)$
 ${0.06331207100}{+}{0.}{}{I}$ (4)
 > $\mathrm{evalf}\left(\mathrm{ans},20\right)$
 ${0.063312071018173992738}{+}{0.}{}{I}$ (5)

Trig form

 > $\mathrm{int}\left(\mathrm{sqrt}\left(1+2\mathrm{sin}\left(x\right)\right),x=0..\frac{\mathrm{\pi }}{2}\right)$
 ${-}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{3}}}{{2}}\right){+}{\mathrm{EllipticF}}{}\left(\frac{\sqrt{{2}}{}\sqrt{{3}}}{{3}}{,}\frac{\sqrt{{3}}}{{2}}\right){+}{4}{}{\mathrm{EllipticE}}{}\left(\frac{\sqrt{{3}}}{{2}}\right){-}{\mathrm{EllipticPi}}{}\left(\frac{\sqrt{{2}}{}\sqrt{{3}}}{{3}}{,}\frac{{3}}{{4}}{,}\frac{\sqrt{{3}}}{{2}}\right)$ (6)

Indefinite trig form

 > $\mathrm{Itrig}≔\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(1+2\mathrm{cos}\left(x\right)\right)},x\right)$
 ${\mathrm{Itrig}}{≔}\frac{{2}{}\sqrt{{3}}{}{\mathrm{InverseJacobiAM}}{}\left(\frac{{x}}{{2}}{,}\frac{{2}{}\sqrt{{3}}}{{3}}\right)}{{3}}$ (7)

Check answer:

 > $\mathrm{simplify}\left(\mathrm{combine}\left(\mathrm{diff}\left(\mathrm{Itrig},x\right)-\frac{1}{\mathrm{sqrt}\left(1+2\mathrm{cos}\left(x\right)\right)},\mathrm{trig}\right)\right)$
 ${0}$ (8)

References

 Labahn, G., and Mutrie, M. "Reduction of Elliptic Integrals to Legendre Normal Form." University of Waterloo Tech Report 97-21, Department of Computer Science, 1997.

 See Also