Symbolics - Maple Help

Enhancements to Symbolic Capabilities in Maple 13

Maple 13 includes enhancements to its symbolic capabilities in the following areas.

New calling sequences for int

 • Multiple definite integrals can be computed with a single call to int using a list or a sequence.  A list and sequence are generally treated the same, but a list is safer for indefinite integrals to make sure variables are not confused for other integration options.
 > int(x+y, [x=0..1, y=0..1]);
 ${1}$ (1)
 > int(x+y, x=0..1, y=0..1);
 ${1}$ (2)
 > int((x,y)->x+y, [0..1, 0..1]);
 ${1}$ (3)
 > int((x,y)->x+y, 0..1, 0..1);
 ${1}$ (4)
 • Multiple indefinite integrals can also be computed using a similar notation to diff
 > diff(int(x+y, x, y), x, y);
 ${x}{+}{y}$ (5)
 > int(x+y, [x,y]);
 $\frac{{1}}{{2}}{}{{x}}^{{2}}{}{y}{+}\frac{{1}}{{2}}{}{{y}}^{{2}}{}{x}$ (6)
 • Ability to select internal integration routines with the method option. See int,methods for more information about available methods.
 > int(1/sqrt((1-t^2)*(1-2*t^2)), t=0..1, method=FTOC);
 ${\mathrm{EllipticK}}{}\left(\sqrt{{2}}\right)$ (7)
 > int(1/sqrt((1-t^2)*(1-2*t^2)), t=0..1, method=Elliptic);
 ${-}\frac{{I}{}\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}{+}\frac{\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}$ (8)
 • If a method does not return a result, an unevaluated int call will be returned with the given method option.
 > int(1/sqrt((1-t^2)*(1-2*t^2)), t=0..1, method=Polynomial);
 ${\mathrm{int}}{}\left(\frac{{1}}{\sqrt{\left({-}{{t}}^{{2}}{+}{1}\right){}\left({-}{2}{}{{t}}^{{2}}{+}{1}\right)}}{,}{t}{=}{0}{..}{1}{,}{\mathrm{method}}{=}{\mathrm{Polynomial}}\right)$ (9)
 • There is also a new option to int to perform numeric integration.  See Numerics Updates in Maple 13 for more information.

Improved Integration of erf, Ci, and Si functions

 • Maple 13 includes improved handling of integrals involving erf, Ci, Si, FresnelS, and FresnelC integrals.
 • The following definite integrals could not be computed by previous versions of Maple.
 > int(x*erf(x)*exp(-x^2), x);
 ${-}\frac{{\mathrm{erf}}{}\left({x}\right){}{{ⅇ}}^{{-}{{x}}^{{2}}}}{{2}}{+}\frac{\sqrt{{2}}{}{\mathrm{erf}}{}\left(\sqrt{{2}}{}{x}\right)}{{4}}$ (10)
 > int(x^3*erf(x)*exp(-x^2), x);
 ${\mathrm{erf}}{}\left({x}\right){}\left({-}\frac{{{x}}^{{2}}}{{2}{}{{ⅇ}}^{{{x}}^{{2}}}}{-}\frac{{1}}{{2}{}{{ⅇ}}^{{{x}}^{{2}}}}\right){-}\frac{{-}\frac{{5}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{erf}}{}\left(\sqrt{{2}}{}{x}\right)}{{16}}{+}\frac{{x}}{{4}{}{\left({{ⅇ}}^{{{x}}^{{2}}}\right)}^{{2}}}}{\sqrt{{\mathrm{\pi }}}}$ (11)
 > int(Ci(x)*sin(x), x);
 ${-}{\mathrm{Ci}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right){+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{2}}{+}\frac{{\mathrm{Ci}}{}\left({2}{}{x}\right)}{{2}}$ (12)
 > int(Si(x)*cos(x), x);
 ${\mathrm{Si}}{}\left({x}\right){}{\mathrm{sin}}{}\left({x}\right){-}\frac{{\mathrm{ln}}{}\left({x}\right)}{{2}}{+}\frac{{\mathrm{Ci}}{}\left({2}{}{x}\right)}{{2}}$ (13)
 > int(FresnelS(2*x)^2, x);
 ${{\mathrm{FresnelS}}{}\left({2}{}{x}\right)}^{{2}}{}{x}{+}\frac{{\mathrm{FresnelS}}{}\left({2}{}{x}\right){}{\mathrm{cos}}{}\left({2}{}{\mathrm{\pi }}{}{{x}}^{{2}}\right)}{{\mathrm{\pi }}}{-}\frac{\sqrt{{2}}{}{\mathrm{FresnelS}}{}\left({2}{}\sqrt{{2}}{}{x}\right)}{{4}{}{\mathrm{\pi }}}$ (14)
 > int(FresnelC(2*x)*cos(2*Pi*x^2)*x, x);
 $\frac{{\mathrm{FresnelC}}{}\left({2}{}{x}\right){}{\mathrm{sin}}{}\left({2}{}{\mathrm{\pi }}{}{{x}}^{{2}}\right)}{{4}{}{\mathrm{\pi }}}{-}\frac{\sqrt{{2}}{}{\mathrm{FresnelS}}{}\left({2}{}\sqrt{{2}}{}{x}\right)}{{16}{}{\mathrm{\pi }}}$ (15)

Extended functionality of lcoeff, tcoeff, and sort

 • Support for the monomial orders plex, grlex, and tdeg was added to these three commands. They can be specified using a second argument of the form order=mo(x,y,...), where mo is one of the three names above and x, y, ... are variable names. For more details, see lcoeff and sort.
 > f := 4*x^3 + 5*x^2*z^2 + 2*x*y^2*z + 1;
 ${f}{≔}{5}{}{{x}}^{{2}}{}{{z}}^{{2}}{+}{2}{}{x}{}{{y}}^{{2}}{}{z}{+}{4}{}{{x}}^{{3}}{+}{1}$ (16)
 > (lcoeff,tcoeff)(f, order=grlex(x,y,z));
 ${5}{,}{1}$ (17)
 > sort(f, order=tdeg(x,y,z));
 ${2}{}{x}{}{{y}}^{{2}}{}{z}{+}{5}{}{{x}}^{{2}}{}{{z}}^{{2}}{+}{4}{}{{x}}^{{3}}{+}{1}$ (18)
 • The efficiency of lcoeff and tcoeff was improved as well. For information, see Efficiency Improvements in Maple 13.

Solve Enhancements

 • solve will accept as options all of the environment variables that affect its behavior.  See solve,details for a complete list.
 > solve(x^4-1);
 ${1}{,}{-1}{,}{I}{,}{-I}$ (19)
 > solve(x^4-1,MaxSols=1);
 ${1}$ (20)
 • solve now accepts the new option UseAssumptions which will cause solve to process any assumptions on the independent variables into additional equations and inequations.
 > solve(x^4-1, 'UseAssumptions') assuming x>0;
 ${1}$ (21)
 > solve(x^4-1, 'UseAssumptions') assuming abs(x)<>1;
 • solve will now return solutions to inequalities containing trig and arctrig functions.
 > solve(sin(x)>0);
 $\left({-}{2}{}{\mathrm{\pi }}{,}{-}{\mathrm{\pi }}\right){,}\left({0}{,}{\mathrm{\pi }}\right)$ (22)
 > solve(arcsin(1/x)>0);
 $\left[{1}{,}{\mathrm{\infty }}\right)$ (23)
 > solve(arcsin(1/x)*sin(x-3)*cos(x+3)>0);
 $\left({-}{3}{-}\frac{{3}{}{\mathrm{\pi }}}{{2}}{,}{-}{3}{}{\mathrm{\pi }}{+}{3}\right){,}\left({-}{3}{-}\frac{{\mathrm{\pi }}}{{2}}{,}{-}{2}{}{\mathrm{\pi }}{+}{3}\right){,}\left({-}{3}{+}\frac{{\mathrm{\pi }}}{{2}}{,}{-1}\right]{,}\left[{1}{,}{-}{3}{+}\frac{{3}{}{\mathrm{\pi }}}{{2}}\right){,}\left({3}{,}{-}{3}{+}\frac{{5}{}{\mathrm{\pi }}}{{2}}\right){,}\left({\mathrm{\pi }}{+}{3}{,}{-}{3}{+}\frac{{7}{}{\mathrm{\pi }}}{{2}}\right)$ (24)
 • RealDomain:-solve uses the new SemiAlgebraicSetTools subpackage of RegularChains to solve systems of polynomials and rational functions which have finitely many solutions.  This allows it to produce much better solutions than those produced by previous versions of Maple.
 > RealDomain:-solve({x^9+x+y,1/y^2-1},{x,y});
 $\left\{{x}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{9}}{+}{\mathrm{_Z}}{-}{1}{,}{0.8242998123}{..}{0.824300766}\right){,}{y}{=}{-1}\right\}{,}\left\{{x}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{9}}{+}{\mathrm{_Z}}{+}{1}{,}{-0.824300766}{..}{-0.8242998123}\right){,}{y}{=}{1}\right\}$ (25)

Improved performance in evala/Gcd

 The efficiency of the evala/Gcd command has been improved. See Efficiency Improvements in Maple 13.

Improved performance in mod/Expand and mod/Divide

 The efficiency of the Expand and Divide commands for modular multivariate polynomial expansion and division has been improved. See Efficiency Improvements in Maple 13.