Elliptic Integrals - Maple Programming Help

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Elliptic Integrals

Description

 • Elliptic integrals are integrals of the form

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

 • 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

 > ${∫}_{0}^{\frac{1}{2}}\frac{\sqrt{1+{x}^{4}}}{1-{x}^{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${-}\frac{{1}}{{8}}{}\sqrt{{2}}{}{\mathrm{ln}}{}\left(\sqrt{{17}}{-}{2}{}\sqrt{{2}}\right){+}\frac{{1}}{{8}}{}\sqrt{{2}}{}{\mathrm{ln}}{}\left(\sqrt{{17}}{+}{2}{}\sqrt{{2}}\right){-}\frac{{1}}{{4}}{}\sqrt{{2}}{}{\mathrm{arctan}}{}\left(\frac{{1}}{{4}}{}\sqrt{{17}}{}\sqrt{{2}}\right){+}\frac{{1}}{{8}}{}{\mathrm{π}}{}\sqrt{{2}}$ (1)

Symbolic parameters

 > $\mathrm{assume}\left(0
 > ${∫}_{0}^{k}\frac{{x}^{2}}{\sqrt{\left(1-{x}^{2}\right)\left(1-{k}^{2}{x}^{2}\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 $\frac{{\mathrm{EllipticF}}{}\left({\mathrm{k~}}{,}{\mathrm{k~}}\right)}{{{\mathrm{k~}}}^{{2}}}{-}\frac{{\mathrm{EllipticE}}{}\left({\mathrm{k~}}{,}{\mathrm{k~}}\right)}{{{\mathrm{k~}}}^{{2}}}$ (2)

 > $\mathrm{ans}≔{∫}_{0}^{\frac{1}{4}}\frac{1}{\left({x}^{4}+2\right)\sqrt{4-5{x}^{2}+{x}^{4}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${\mathrm{ans}}{≔}\frac{{1}}{{16}}{}{\sum }_{{\mathrm{_α}}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{4}}{+}{2}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\mathrm{EllipticPi}}{}\left(\frac{{1}}{{4}}{,}\frac{{1}}{{{\mathrm{_α}}}^{{2}}}{,}\frac{{1}}{{2}}\right)$ (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

 > ${∫}_{0}^{\frac{\mathrm{π}}{2}}\sqrt{1+2\mathrm{sin}\left(x\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${-}{\mathrm{EllipticK}}{}\left(\frac{{1}}{{2}}{}\sqrt{{3}}\right){+}{\mathrm{EllipticF}}{}\left(\frac{{1}}{{3}}{}\sqrt{{2}}{}\sqrt{{3}}{,}\frac{{1}}{{2}}{}\sqrt{{3}}\right){+}{4}{}{\mathrm{EllipticE}}{}\left(\frac{{1}}{{2}}{}\sqrt{{3}}\right){-}{\mathrm{EllipticPi}}{}\left(\frac{{1}}{{3}}{}\sqrt{{2}}{}\sqrt{{3}}{,}\frac{{3}}{{4}}{,}\frac{{1}}{{2}}{}\sqrt{{3}}\right)$ (6)

Indefinite trig form

 > $\mathrm{Itrig}≔∫\frac{1}{\sqrt{1+2\mathrm{cos}\left(x\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${\mathrm{Itrig}}{≔}\frac{{2}}{{3}}{}\sqrt{{3}}{}{\mathrm{InverseJacobiAM}}{}\left(\frac{{1}}{{2}}{}{x}{,}\frac{{2}}{{3}}{}\sqrt{{3}}\right)$ (7)

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