Symbolics - Maple Help

Enhancements to Symbolic Capabilities in Maple 14

Maple 14 includes enhancements to its symbolic capabilities in the following areas.

The assuming Command

 • Numerous improvements have been made to the assuming command.
 • The assuming command now handles assumptions of the form "for all variables" given together with assumptions of the form "only for this variable", providing regarded additional flexibility when experimenting with results under varied assumptions. Also, in Maple 14 assuming works also with equation labels
 > sqrt((y-x)^2);
 $\sqrt{{\left({y}{-}{x}\right)}^{{2}}}$ (1)
 > simplify((1)) assuming positive and x < y;
 ${y}{-}{x}$ (2)

A more complicated example

 > [about(x), about(y)] assuming x, posint, odd, x < 10, y, negint, even, y > -10;
 Originally x, renamed x~:   is assumed to be: AndProp(LinearProp(2,integer,1),RealRange(1,9)) Originally y, renamed y~:   is assumed to be: AndProp(LinearProp(2,integer,0),RealRange(-8,-2))
 $\left[\right]$ (3)

The assumption placed above on $x$ and $y$ is equivalent to $x::\left({ℤ}^{+}\wedge \mathrm{odd}\right),x<10,y::\left({ℤ}^{-}\wedge \mathrm{even}\right),-10 and, as usual, it is a temporary assumption, so that without assuming nothing is known about(x) or about(y).

 • The assuming command now handles the optional arguments of Matrix and related commands; for illustration, in this example the optional argument is readonly = true
 > simplify(Matrix([[sqrt(x^2), 0], [0, sqrt(y^2)]], readonly = true)) assuming positive;
 $\left[\begin{array}{cc}{x}& {0}\\ {0}& {y}\end{array}\right]$ (4)
 • The assuming command now avoids placing assumptions in all Maple mappings or programs that implement their options using the modern keyword parameters approach. Compare for instance these two results
 > a*x^2+b*x+c;
 ${a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}$ (5)
 > SolveTools:-Polynomial((5), x, domain = parametric);
 $\left\{\begin{array}{cc}\left\{\begin{array}{cc}\left\{\begin{array}{cc}\left[{x}\right]& {c}{=}{0}\\ \left[\right]& {\mathrm{otherwise}}\end{array}\right\& {b}{=}{0}\\ \left[{-}\frac{{c}}{{b}}\right]& {\mathrm{otherwise}}\end{array}\right\& {a}{=}{0}\\ \left[\frac{{-}{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}{,}{-}\frac{{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}\right]& {\mathrm{otherwise}}\end{array}\right\$ (6)
 > SolveTools:-Polynomial((5), x, domain = parametric) assuming Not(0);
 $\left[\frac{{-}{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}{,}{-}\frac{{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}\right]$ (7)
 • Compositions of map and @ are now also possible when using assuming, for example
 > [(y^2-2*y*x+x^2)^(1/2), (y^2+2*y*x+x^2)^(1/2)];
 $\left[\sqrt{{{x}}^{{2}}{-}{2}{}{y}{}{x}{+}{{y}}^{{2}}}{,}\sqrt{{{x}}^{{2}}{+}{2}{}{y}{}{x}{+}{{y}}^{{2}}}\right]$ (8)
 > map(simplify@factor, (8)) assuming real, y > x;
 $\left[{y}{-}{x}{,}\left|{x}{+}{y}\right|\right]$ (9)

 • More formulas were added to the conversion network for mathematical functions. See convert for a full list of function conversions in Maple.
 > convert(AngerJ(a,z), Bessel) assuming a::integer;
 ${\mathrm{BesselJ}}{}\left({a}{,}{z}\right)$ (10)
 $\left[{\mathrm{arctan}}{}\left({y}{,}{x}\right){=}\frac{\sqrt{\frac{{\left({-}{I}{}{x}{+}{y}\right)}^{{2}}}{{{x}}^{{2}}{+}{{y}}^{{2}}}}{}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}{}\sqrt{\frac{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{-}{1}}{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{+}{1}}}{}\sqrt{\frac{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{+}{1}}{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{-}{1}}}{}{\mathrm{arccos}}{}\left(\frac{{x}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}\right)}{{-}{I}{}{x}{+}{y}}{+}{\mathrm{\pi }}{-}\frac{{\mathrm{\pi }}{}\sqrt{{-}\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{-}{1}}{}\sqrt{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}}}{\sqrt{{-}\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}}{}\sqrt{\frac{{x}{+}{I}{}{y}}{\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}}{+}{1}}}{,}{\mathrm{with no restrictions on}}{}\left({y}{,}{x}\right)\right]$ (11)

Improved Symbolic Order Differentiation

Maple 14 includes expanded differentiation knowledge for symbolic order derivatives ($\frac{{d}^{n}}{{\mathrm{dz}}^{n}}$ with symbolic $n$)

 • Powers
 > (a+z)^lambda;
 ${\left({a}{+}{z}\right)}^{{\mathrm{\lambda }}}$ (12)
 > %diff((12), z$n) = diff((12), z$n);
 $\frac{{{\partial }}^{{n}}}{{\partial }{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\left({a}{+}{z}\right)}^{{\mathrm{\lambda }}}\right){=}{\mathrm{pochhammer}}{}\left({\mathrm{\lambda }}{-}{n}{+}{1}{,}{n}\right){}{\left({a}{+}{z}\right)}^{{\mathrm{\lambda }}{-}{n}}$ (13)
 > 1/(a+b*z^3);
 $\frac{{1}}{{b}{}{{z}}^{{3}}{+}{a}}$ (14)
 > %diff((14), z$n) = diff((14), z$n);
 $\frac{{{\partial }}^{{n}}}{{\partial }{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{1}}{{b}{}{{z}}^{{3}}{+}{a}}\right){=}{\sum }_{{\mathrm{_α}}{=}{\mathrm{RootOf}}{}\left({b}{}{{\mathrm{_Z}}}^{{3}}{+}{a}\right)}{}\frac{{\mathrm{pochhammer}}{}\left({-}{n}{,}{n}\right){}{\left({z}{-}{\mathrm{_α}}\right)}^{{-}{1}{-}{n}}}{{3}{}{b}{}{{\mathrm{_α}}}^{{2}}}$ (15)
 • Elementary functions
 > tan(z);
 ${\mathrm{tan}}{}\left({z}\right)$ (16)
 > %diff((16), z$n) = diff((16), z$n);
 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{tan}}{}\left({z}\right){=}{-}{{I}}^{{n}{+}{1}}{}{{2}}^{{n}}{}\left({I}{}{\mathrm{tan}}{}\left({z}\right){-}{1}{+}\left(\left\{\begin{array}{cc}{1}& {n}{=}{0}\\ {0}& {\mathrm{otherwise}}\end{array}\right\\right)\right){}\left({\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}}{}\frac{{\left({-1}\right)}^{{\mathrm{_k1}}}{}{\mathrm{_k1}}{!}{}{\mathrm{Stirling2}}{}\left({n}{,}{\mathrm{_k1}}\right){}{\left({I}{}{\mathrm{tan}}{}\left({z}\right){+}{1}\right)}^{{\mathrm{_k1}}}}{{{2}}^{{\mathrm{_k1}}}}\right)$ (17)
 > csch(z);
 ${\mathrm{csch}}{}\left({z}\right)$ (18)
 > %diff((18), z$n) = diff((18), z$n);
 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{csch}}{}\left({z}\right){=}{\left({{I}}^{{n}}\right)}^{{2}}{}{\mathrm{csch}}{}\left({z}\right){}\left({\sum }_{{\mathrm{_j1}}{=}{0}}^{{n}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{_j1}}}{}{\left({-1}\right)}^{{\mathrm{_k1}}}{}\left(\genfrac{}{}{0}{}{{n}}{{\mathrm{_j1}}}\right){}{{2}}^{{\mathrm{_j1}}{-}{\mathrm{_k1}}}{}{\mathrm{_k1}}{!}{}{\mathrm{Stirling2}}{}\left({\mathrm{_j1}}{,}{\mathrm{_k1}}\right){}{\left({1}{-}{\mathrm{coth}}{}\left({z}\right)\right)}^{{\mathrm{_k1}}}\right)$ (19)
 • Special kinds of Bessel functions
 > AngerJ(n,z);
 ${\mathrm{AngerJ}}{}\left({n}{,}{z}\right)$ (20)
 > %diff((20), z$n) = diff((20), z$n);
 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{AngerJ}}{}\left({n}{,}{z}\right){=}\frac{{\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}}{}{\left({-1}\right)}^{{\mathrm{_k1}}}{}\left(\genfrac{}{}{0}{}{{n}}{{\mathrm{_k1}}}\right){}{\mathrm{BesselJ}}{}\left({2}{}{\mathrm{_k1}}{,}{z}\right)}{{{2}}^{{n}}}$ (21)
 > HankelH1(n,z);
 ${\mathrm{HankelH1}}{}\left({n}{,}{z}\right)$ (22)
 > %diff((22), z$n) = diff((22), z$n);
 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{HankelH1}}{}\left({n}{,}{z}\right){=}\frac{{\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}}{}{\left({-1}\right)}^{{\mathrm{_k1}}}{}\left(\genfrac{}{}{0}{}{{n}}{{\mathrm{_k1}}}\right){}{\mathrm{BesselJ}}{}\left({2}{}{\mathrm{_k1}}{,}{z}\right)}{{{2}}^{{n}}}{+}\frac{{I}{}\left({\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}}{}{\left({-1}\right)}^{{\mathrm{_k1}}}{}\left(\genfrac{}{}{0}{}{{n}}{{\mathrm{_k1}}}\right){}{\mathrm{BesselY}}{}\left({2}{}{\mathrm{_k1}}{,}{z}\right)\right)}{{{2}}^{{n}}}$ (23)
 • Confluent type of hypergeometric functions
 > CylinderD(a,z);
 ${\mathrm{CylinderD}}{}\left({a}{,}{z}\right)$ (24)
 > %diff((24), z$n) = diff((24), z$n);
 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{CylinderD}}{}\left({a}{,}{z}\right){=}\frac{{\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{n}}{{\mathrm{_k1}}}\right){}\left({\sum }_{{\mathrm{_k3}}{=}{0}}^{{\mathrm{_k1}}}{}{{ⅇ}}^{{-}\frac{{{z}}^{{2}}}{{4}}}{}{\mathrm{IncompleteBellB}}{}\left({\mathrm{_k1}}{,}{\mathrm{_k3}}{,}{-}\frac{{\mathrm{pochhammer}}{}\left({2}{,}{1}\right){}{z}}{{4}}{,}{\mathrm{...}}{,}{-}\frac{{\mathrm{pochhammer}}{}\left({2}{-}{\mathrm{_k1}}{+}{\mathrm{_k3}}{,}{\mathrm{_k1}}{-}{\mathrm{_k3}}{+}{1}\right){}{{z}}^{{1}{-}{\mathrm{_k1}}{+}{\mathrm{_k3}}}}{{4}}\right)\right){}{{2}}^{{n}{-}{\mathrm{_k1}}}{}{a}{!}{}{\mathrm{HermiteH}}{}\left({a}{-}{n}{+}{\mathrm{_k1}}{,}\frac{{z}{}\sqrt{{2}}}{{2}}\right){}{\left(\frac{\sqrt{{2}}}{{2}}\right)}^{{n}{-}{\mathrm{_k1}}}}{\left({a}{-}{n}{+}{\mathrm{_k1}}\right){!}}}{{{2}}^{\frac{{a}}{{2}}}}$ (25)

New Solve Command in PDEtools

 • Solve is a new unified command of the PDEtools package to compute exact, series or numerical solutions, possibly independent of indicated variables, for systems of algebraic or differential equations, possibly including inequations, initial values or boundary conditions. In this sense Solve provides the functionality of solve, fsolve, dsolve, pdsolve and more, through a single command that understands which command is to be called according to your input. Second, Solve has the ability to compute these solutions independent of specified variables, generalizing the functionality provided by solve[identity] to exact, series or numerical solutions, to systems of non-polynomial algebraic and/or differential equations, and allowing for an arbitrary number of identity variables. This unification together with its new solving capability make concretely simpler the interactive study of the solutions of an algebraic or differential equation system.
 Examples
 > with(PDEtools, Solve);
 $\left[{\mathrm{Solve}}\right]$ (26)
 A non-differential equation
 > eq[1] := a*x^2 + b*x + c;
 ${{\mathrm{eq}}}_{{1}}{≔}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}$ (27)
 > Solve(eq[1], x);
 ${x}{=}\frac{{-}{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}{,}{x}{=}{-}\frac{{b}{+}\sqrt{{-}{4}{}{a}{}{c}{+}{{b}}^{{2}}}}{{2}{}{a}}$ (28)
 An ODE problem and its series solution
 > eq[2] := diff(y(x),x) = y(x);
 ${{\mathrm{eq}}}_{{2}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{y}{}\left({x}\right)$ (29)
 > Solve(eq[2], series);
 ${y}{}\left({x}\right){=}{y}{}\left({0}\right){+}{y}{}\left({0}\right){}{x}{+}\frac{{1}}{{2}}{}{y}{}\left({0}\right){}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{y}{}\left({0}\right){}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{y}{}\left({0}\right){}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{y}{}\left({0}\right){}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (30)
 A PDE problem with boundary conditions
 > eq[3] := [diff(u(x, t), t)+c*(diff(u(x, t), x)) = -lambda*u(x, t), u(x, 0) = phi(x)];
 ${{\mathrm{eq}}}_{{3}}{≔}\left[\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){+}{c}{}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right){=}{-}{\mathrm{\lambda }}{}{u}{}\left({x}{,}{t}\right){,}{u}{}\left({x}{,}{0}\right){=}{\mathrm{\phi }}{}\left({x}\right)\right]$ (31)
 > Solve(eq[3]);
 $\left\{{u}{}\left({x}{,}{t}\right){=}{\mathrm{\phi }}{}\left({-}{t}{}{c}{+}{x}\right){}{{ⅇ}}^{{-}{\mathrm{\lambda }}{}{t}}\right\}$ (32)
 Numerical solution for a PDE with boundary conditions
 > eq[4] := [diff(u(x,t),t) = -diff(u(x,t),x), u(x,0) = sin(2*Pi*x), u(0,t) = -sin(2*Pi*t)];
 ${{\mathrm{eq}}}_{{4}}{≔}\left[\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){=}{-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){,}{u}{}\left({x}{,}{0}\right){=}{\mathrm{sin}}{}\left({2}{}{\mathrm{\pi }}{}{x}\right){,}{u}{}\left({0}{,}{t}\right){=}{-}{\mathrm{sin}}{}\left({2}{}{\mathrm{\pi }}{}{t}\right)\right]$ (33)
 > sol[4] := PDEtools:-Solve(eq[4], numeric, time=t, range=0..1);
 ${{\mathrm{sol}}}_{{4}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{export}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{plot}}{,}{\mathrm{plot3d}}{,}{\mathrm{animate}}{,}{\mathrm{value}}{,}{\mathrm{settings}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (34)
 > sol[4]:-plot(t=0, numpoints=50);
 In the examples above the advantage with regards to calling solve, dsolve or pdsolve is in using a single command and having a unified format for the input and output. Solve however also provides additional functionality: it can compute solutions independent of indicated variables. The system being solved using independentof can also contain inequations
 > eq[5] := [k*a*c*(a+b)*exp(k*d*t)-2*a*exp(k*t)*k+Q*(-c+a)*x, a <> 0];
 ${{\mathrm{eq}}}_{{5}}{≔}\left[{k}{}{a}{}{c}{}\left({a}{+}{b}\right){}{{ⅇ}}^{{k}{}{d}{}{t}}{-}{2}{}{a}{}{{ⅇ}}^{{k}{}{t}}{}{k}{+}{Q}{}\left({-}{c}{+}{a}\right){}{x}{,}{a}{\ne }{0}\right]$ (35)
 > Solve(eq[5], {a, b, c, d}, independentof = {t, x});
 $\left\{{a}{=}{a}{,}{b}{=}{-}\frac{{{a}}^{{2}}{-}{2}}{{a}}{,}{c}{=}{a}{,}{d}{=}{1}\right\}$ (36)
 Solutions that are independent of the specified variables can be computed as well for differential equations or systems of them;
 > eq[6] := diff(f(x,y),x)*diff(g(x,y),x) + diff(f(x,y),y)*diff(g(x,y),y) + g(x,y)*(diff(f(x,y), x,x) + diff(f(x,y), y,y)) = -1;
 ${{\mathrm{eq}}}_{{6}}{≔}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}{,}{y}\right)\right){+}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({x}{,}{y}\right)\right){+}{g}{}\left({x}{,}{y}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){=}{-1}$ (37)
 Here are solutions for this PDE that are independent of $x$ and independent of $y$
 > Solve(eq[6], independentof = x);
 $\left\{{f}{}\left({x}{,}{y}\right){=}\mathrm{f__1}{}\left({y}\right){,}{g}{}\left({x}{,}{y}\right){=}\frac{{-}{y}{+}\mathrm{c__1}}{\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{f__1}{}\left({y}\right)}\right\}$ (38)
 > Solve(eq[6], independentof = y);
 $\left\{{f}{}\left({x}{,}{y}\right){=}\mathrm{f__1}{}\left({x}\right){,}{g}{}\left({x}{,}{y}\right){=}\frac{{-}{x}{+}\mathrm{c__1}}{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{f__1}{}\left({x}\right)}\right\}$ (39)

Algebraic Solver

 • Users have improved access to the polynomial solvers in the SolveTools package through a command, SolveTools[Polynomial], and many new options. Univariate polynomial solving can be accessed directly with the new command SolveTools[Polynomial] and multivariate solving can be accessed with SolveTools[PolynomialSystem].  Both commands can be more efficient than solve on purely polynomial equations, since they avoid a large amount of preprocessing and dispatch overhead much like SolveTools[Linear] does for linear system solving.
 > with(SolveTools):
 > f1 := expand((x-1)^4*eval(z^4-z-1,z=x^3+x)):
 > Polynomial(f1, x, domain=integer);
 $\left[{1}{,}{1}{,}{1}{,}{1}\right]$ (40)
 > Polynomial(f1, x, domain=real);
 $\left[{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{12}}{+}{4}{}{{\mathrm{_Z}}}^{{10}}{+}{6}{}{{\mathrm{_Z}}}^{{8}}{+}{4}{}{{\mathrm{_Z}}}^{{6}}{+}{{\mathrm{_Z}}}^{{4}}{-}{{\mathrm{_Z}}}^{{3}}{-}{\mathrm{_Z}}{-}{1}{,}{-0.5542396980}\right){,}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{12}}{+}{4}{}{{\mathrm{_Z}}}^{{10}}{+}{6}{}{{\mathrm{_Z}}}^{{8}}{+}{4}{}{{\mathrm{_Z}}}^{{6}}{+}{{\mathrm{_Z}}}^{{4}}{-}{{\mathrm{_Z}}}^{{3}}{-}{\mathrm{_Z}}{-}{1}{,}{0.7679130647}\right){,}{1}{,}{1}{,}{1}{,}{1}\right]$ (41)
 • Find piecewise solutions to parametric polynomial equations with SolveTools[Polynomial].
 > Polynomial(a*x^2-(b+a)*x+b, x, domain=parametric);
 $\left\{\begin{array}{cc}\left\{\begin{array}{cc}\left[{x}\right]& {b}{=}{0}\\ \left[{1}\right]& {\mathrm{otherwise}}\end{array}\right\& {a}{=}{0}\\ \left[{1}{,}\frac{{b}}{{a}}\right]& {\mathrm{otherwise}}\end{array}\right\$ (42)
 > PolynomialSystem({y^3+1,y+x^2-1}, {x,y}, domain=real);
 $\left\{{x}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}{,}{-1.414214135}{..}{-1.41421318}\right){,}{y}{=}{-1}\right\}{,}\left\{{x}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}{,}{1.41421318}{..}{1.414214135}\right){,}{y}{=}{-1}\right\}$ (43)
 > PolynomialSystem({y^3+1,y+x^2-1}, {x,y}, domain=real, explicit);
 $\left\{{x}{=}{-}\sqrt{{2}}{,}{y}{=}{-1}\right\}{,}\left\{{x}{=}\sqrt{{2}}{,}{y}{=}{-1}\right\}$ (44)
 > PolynomialSystem({y-x^2+1, y+x^2-1}, [x,y], engine=groebner, backsubstitute=false);
 $\left[\left\{{y}{,}{x}{-}{1}\right\}{,}{\varnothing }\right]{,}\left[\left\{{y}{,}{x}{+}{1}\right\}{,}{\varnothing }\right]$ (45)
 • A new lower level interface has been added to solve via the command SolveTools[Engine].  This interface uses minimal pre- and post-processing and uniform input and output formats more appropriate for use inside user programs. solve is still the recommended user level interface to Maple's solving functionality.
 > Engine({x^2+1},{x});
 $\left[\left\{{x}{=}{-I}\right\}{,}\left\{{x}{=}{I}\right\}\right]$ (46)
 • The allvalues command has been changed to use the RootFinding library on non-polynomial RootOf inputs allowing it to return more comprehensive answers.
 > solve(x*sin(x)-1);
 ${\mathrm{RootOf}}{}\left({\mathrm{_Z}}{}{\mathrm{sin}}{}\left({\mathrm{_Z}}\right){-}{1}\right)$ (47)
 > allvalues((47));
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