 Advanced Math - Maple Programming Help

Maple 2019 includes numerous cutting-edge updates in a variety of branches of mathematics.

Differential Equations

The new command FindODE, in the DEtools package, tries to find a linear ordinary differential equation with polynomial coefficients for the given expression.

 > $\mathrm{DEtools}\left[\mathrm{FindODE}\right]\left(\mathrm{cos}\left(\mathrm{sqrt}\left(x\right)\right),y\left(x\right)\right)$
 ${y}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1.1)
 >
 $\left({-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{-}{4}{}{{x}}^{{2}}{-}{3}{}{x}\right){}{y}{}\left({x}\right){+}\left({-}{{x}}^{{3}}{+}{x}{+}{4}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{2}{}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1.2)

int

Description

 • There have been various improvements made to the int command for Maple 2019.

int Examples

 • New results from int:
 > ${\int }_{0}^{\mathrm{\pi }}\left|\mathrm{sin}\left({x}^{2}\right)\right|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
 ${-}\frac{{\mathrm{FresnelS}}{}\left(\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}\right){}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{2}}{+}{\mathrm{FresnelS}}{}\left(\sqrt{{2}}\right){}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{-}{\mathrm{FresnelS}}{}\left({2}\right){}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{+}{\mathrm{FresnelS}}{}\left(\sqrt{{3}}{}\sqrt{{2}}\right){}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}$ (2.2.1)
 > ${\int }_{0}^{\mathrm{\pi }}{ⅇ}^{I\mathrm{sin}\left(x\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${\mathrm{\pi }}{}{\mathrm{BesselJ}}{}\left({0}{,}{1}\right){+}{I}{}{\mathrm{\pi }}{}{\mathrm{StruveH}}{}\left({0}{,}{1}\right)$ (2.2.2)
 > ${\int }_{0}^{\frac{\mathrm{\pi }}{2}}{\mathrm{sec}\left(u\right)}^{2}{ⅇ}^{-\mathrm{sec}\left(u\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆu$
 ${\mathrm{BesselK}}{}\left({1}{,}{1}\right)$ (2.2.3)
 > ${\int }_{0}^{1}\mathrm{signum}\left(\mathrm{arccos}\left(\frac{1}{z}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆz$
 ${I}$ (2.2.4)
 >
 ${-}\frac{{2}{}\left({\mathrm{sin}}{}\left({\mathrm{\pi }}{}{p}\right){}{\mathrm{signum}}{}\left({\mathrm{sin}}{}\left({\mathrm{\pi }}{}{p}\right)\right){}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{p}}\right){-}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{p}\right){}{\mathrm{signum}}{}\left({\mathrm{sin}}{}\left({\mathrm{\pi }}{}{p}\right)\right){-}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}\left(⌊{p}⌋{+}{1}\right)}{{p}}\right){}{p}{+}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{p}}\right){}{p}{+}{p}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}⌊{p}⌋}{{p}}\right)\right)}{{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{p}}\right){}{{p}}^{{2}}{-}{{p}}^{{2}}{-}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{p}}\right){+}{1}}$ (2.2.5)
 > ${\int }_{0}^{\frac{\mathrm{\pi }}{2}}{\left(\mathrm{csc}\left(x\right)-1\right)}^{\mathrm{\nu }}\mathrm{cos}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 $\frac{{\mathrm{\pi }}{}{\mathrm{\nu }}}{{\mathrm{sin}}{}\left({\mathrm{\nu }}{}{\mathrm{\pi }}\right)}$ (2.2.6)
 > ${\int }_{0}^{2}\left({x}^{2}-\frac{1}{2}+\frac{I\mathrm{ln}\left(-{ⅇ}^{2I\mathrm{\pi }{x}^{2}}\right)}{2\mathrm{\pi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${5}{-}\sqrt{{2}}{-}\sqrt{{3}}$ (2.2.7)
 • Improved answers for definite integrals when the AllSolutions option is given:
 >
 $\frac{\left(\left\{\begin{array}{cc}{-}{⌈{x}{-}\frac{{1}}{{2}}⌉}^{{2}}{+}{x}{}\left(\left\{\begin{array}{cc}{2}{}{x}{-}{1}& \left({x}{+}\frac{{1}}{{2}}\right){::}{'}{\mathrm{integer}}{'}\\ {2}{}⌊{x}⌉& {\mathrm{otherwise}}\end{array}\right\\right)& {0}{<}{x}\\ {-}{⌊{x}{+}\frac{{1}}{{2}}⌋}^{{2}}{+}{x}{}\left(\left\{\begin{array}{cc}{2}{}{x}{+}{1}& \left({x}{-}\frac{{1}}{{2}}\right){::}{'}{\mathrm{integer}}{'}\\ {2}{}⌊{x}⌉& {\mathrm{otherwise}}\end{array}\right\\right)& {x}{\le }{0}\end{array}\right\\right)}{{2}}$ (2.2.8)
 > $\mathrm{int}\left(\frac{1}{2+\mathrm{cos}\left(t\right)},t=0..x,\mathrm{AllSolutions}\right)$
 $\frac{\left(\left\{\begin{array}{cc}{2}{}{\mathrm{\pi }}{}⌈{-}\frac{{\mathrm{\pi }}{-}{x}}{{2}{}{\mathrm{\pi }}}⌉{+}\left(\left\{\begin{array}{cc}{\mathrm{\pi }}& \left({-}\frac{{\mathrm{\pi }}{-}{x}}{{2}{}{\mathrm{\pi }}}\right){::}{'}{\mathrm{integer}}{'}\\ {2}{}{\mathrm{arctan}}{}\left(\frac{{\mathrm{tan}}{}\left(\frac{{x}}{{2}}\right){}\sqrt{{3}}}{{3}}\right)& {\mathrm{otherwise}}\end{array}\right\\right)& {0}{<}{x}\\ {2}{}{\mathrm{\pi }}{}⌊{-}\frac{{\mathrm{\pi }}{-}{x}}{{2}{}{\mathrm{\pi }}}⌋{+}{2}{}{\mathrm{\pi }}{-}\left(\left\{\begin{array}{cc}{\mathrm{\pi }}& \left({-}\frac{{\mathrm{\pi }}{-}{x}}{{2}{}{\mathrm{\pi }}}\right){::}{'}{\mathrm{integer}}{'}\\ {-}{2}{}{\mathrm{arctan}}{}\left(\frac{{\mathrm{tan}}{}\left(\frac{{x}}{{2}}\right){}\sqrt{{3}}}{{3}}\right)& {\mathrm{otherwise}}\end{array}\right\\right)& {x}{\le }{0}\end{array}\right\\right){}\sqrt{{3}}}{{3}}$ (2.2.9)
 > $\mathrm{int}\left(\frac{1}{\mathrm{ln}\left(t\right)},t=0..x,\mathrm{AllSolutions},\mathrm{CauchyPrincipalValue}\right)$
 $\left\{\begin{array}{cc}{-}{I}{}{\mathrm{\pi }}{+}{\mathrm{Ei}}{}\left({\mathrm{ln}}{}\left({-}{x}\right){+}{I}{}{\mathrm{\pi }}\right)& {x}{<}{0}\\ {0}& {x}{=}{0}\\ {\mathrm{Ei}}{}\left({\mathrm{ln}}{}\left({x}\right)\right)& {x}{<}{1}\\ {-}{\mathrm{\infty }}& {x}{=}{1}\\ {\mathrm{Ei}}{}\left({\mathrm{ln}}{}\left({x}\right)\right)& {1}{<}{x}\end{array}\right\$ (2.2.10)

Integral Transforms

Description

 • The inttrans package in Maple 2019 has had several transforms, specifically laplace, invlaplace, fourier and invfourier, extended to handle a larger class of problems, and in some cases already handled classes of problems faster. This has been accomplished via an integration by differentiation approach described in the following: - A. Kempf, D.M. Jackson and A.H. Morales, "New Dirac delta function based methods with applications to perturbative expansions in quantum field theory", J. Phys. A:47, 2014 - D. Jia, E. Tang, and A. Kempf, "Integration by differentiation: new proofs, methods and examples", J. Phys. A:50, 2017
 • One can view this approach, in simplest possible terms, as a product rule.

Fourier Examples

 • Here are a few examples which failed to transform in prior versions of Maple, but now transform quite rapidly:
 > $\mathrm{ex1}≔\frac{\frac{{ⅇ}^{-\frac{t}{2}}}{1+{ⅇ}^{-t}}\mathrm{sin}\left(t\right)}{t}$
 ${\mathrm{ex1}}{≔}\frac{{{ⅇ}}^{{-}\frac{{t}}{{2}}}{}{\mathrm{sin}}{}\left({t}\right)}{\left({1}{+}{{ⅇ}}^{{-}{t}}\right){}{t}}$ (3.2.1)
 > $\mathrm{inttrans}\left[\mathrm{fourier}\right]\left(\mathrm{ex1},t,s\right)$
 ${-}\frac{{I}}{{2}}{}{\mathrm{ln}}{}\left(\frac{{\mathrm{csc}}{}\left({I}{}{\mathrm{\pi }}{}{s}{-}\left(\frac{{1}}{{2}}{+}{I}\right){}{\mathrm{\pi }}\right){+}{\mathrm{cot}}{}\left({I}{}{\mathrm{\pi }}{}{s}{-}\left(\frac{{1}}{{2}}{+}{I}\right){}{\mathrm{\pi }}\right)}{{\mathrm{csc}}{}\left({I}{}{\mathrm{\pi }}{}{s}{+}\left({-}\frac{{1}}{{2}}{+}{I}\right){}{\mathrm{\pi }}\right){+}{\mathrm{cot}}{}\left({I}{}{\mathrm{\pi }}{}{s}{+}\left({-}\frac{{1}}{{2}}{+}{I}\right){}{\mathrm{\pi }}\right)}\right)$ (3.2.2)
 > $\mathrm{ex2}≔{ⅇ}^{\frac{t}{2}}\mathrm{ln}\left(1+{ⅇ}^{-t}\right)\mathrm{cos}\left(t\right)$
 ${\mathrm{ex2}}{≔}{{ⅇ}}^{\frac{{t}}{{2}}}{}{\mathrm{ln}}{}\left({1}{+}{{ⅇ}}^{{-}{t}}\right){}{\mathrm{cos}}{}\left({t}\right)$ (3.2.3)
 > $\mathrm{inttrans}\left[\mathrm{fourier}\right]\left(\mathrm{ex2},t,s\right)$
 ${\mathrm{\pi }}{}\left(\frac{{\mathrm{csc}}{}\left({\mathrm{\pi }}{}\left({-}\frac{{1}}{{2}}{+}{I}{}\left({s}{-}{1}\right)\right)\right)}{{2}{}{I}{}{s}{-}{1}{-}{2}{}{I}}{+}\frac{{\mathrm{csc}}{}\left({\mathrm{\pi }}{}\left({-}\frac{{1}}{{2}}{+}{I}{}\left({s}{+}{1}\right)\right)\right)}{{2}{}{I}{}{s}{-}{1}{+}{2}{}{I}}\right)$ (3.2.4)
 > $\mathrm{ex3}≔\frac{{ⅇ}^{-t}\mathrm{Heaviside}\left(t\right)+{ⅇ}^{t}\mathrm{Heaviside}\left(-t\right)}{2\left(t+I\right)}$
 ${\mathrm{ex3}}{≔}\frac{{{ⅇ}}^{{-}{t}}{}{\mathrm{Heaviside}}{}\left({t}\right){+}{{ⅇ}}^{{t}}{}{\mathrm{Heaviside}}{}\left({-}{t}\right)}{{2}{}{t}{+}{2}{}{I}}$ (3.2.5)
 > $\mathrm{inttrans}\left[\mathrm{fourier}\right]\left(\mathrm{ex3},t,s\right)$
 $\frac{{{ⅇ}}^{{-}{s}}{}\left({-}{{\mathrm{Ei}}}_{{1}}{}\left({-}{I}{-}{s}\right){}{{ⅇ}}^{{-I}}{+}{{\mathrm{Ei}}}_{{1}}{}\left({I}{-}{s}\right){}{{ⅇ}}^{{I}}\right)}{{2}}$ (3.2.6)
 > $\mathrm{ex4}≔\frac{\mathrm{csch}\left(t\right)\mathrm{sin}\left(t\right)}{t}$
 ${\mathrm{ex4}}{≔}\frac{{\mathrm{csch}}{}\left({t}\right){}{\mathrm{sin}}{}\left({t}\right)}{{t}}$ (3.2.7)
 > $\mathrm{inttrans}\left[\mathrm{fourier}\right]\left(\mathrm{ex4},t,s\right)$
 ${-I}{}\left({-}{2}{}{\mathrm{\pi }}{+}{\mathrm{ln}}{}\left(\frac{{{ⅇ}}^{{\mathrm{\pi }}{}\left({s}{+}{1}\right)}{+}{1}}{{{ⅇ}}^{{\mathrm{\pi }}{}\left({s}{-}{1}\right)}{+}{1}}\right)\right)$ (3.2.8)

 Multivariate Limits The limit command in Maple 2019 has been enhanced for the case of limits of quotients of multivariate functions. See Multivariate Limits for details.
 Real Roots of Polynomials A new algorithm for univariate polynomials has been added to the RootFinding:-Isolate command. It is particularly efficient for ill-conditioned problems and high accuracy solutions, and it provides certified real root isolation for polynomials with irrational coefficients. See Real Root Finding for details.
 Residue The residue command has a new optional argument that allows the user to specify the maximal order of the underlying series computations. See residue for an example.

simplify

Description

 • The simplify command in Maple 2019 has undergone several improvements, especially with regard to expressions containing piecewise functions.

simplify Examples

 • Simplification of expressions containing piecewise functions has been improved.

Equal, equivalent, or implied piecewise branches are now combined by simplify;

 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}5& x=1\\ 5& x=2\\ 6& \mathrm{otherwise}\end{array}\right\\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
 $\left\{\begin{array}{cc}{5}& {x}{=}{1}{\vee }{x}{=}{2}\\ {6}& {\mathrm{otherwise}}\end{array}\right\$ (7.2.1)
 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}0& \frac{\mathrm{\pi }}{2}=T\\ \mathrm{cos}\left(T\right)& \left(\frac{2T-\mathrm{\pi }}{2\mathrm{\pi }}\right)\colon\colon \mathrm{integer}\\ 1& \mathrm{otherwise}\end{array}\right\\right)$
 $\left\{\begin{array}{cc}{\mathrm{cos}}{}\left({T}\right)& \left({-}\frac{{-}{2}{}{T}{+}{\mathrm{\pi }}}{{2}{}{\mathrm{\pi }}}\right){::}{'}{\mathrm{integer}}{'}\\ {1}& {\mathrm{otherwise}}\end{array}\right\$ (7.2.2)
 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}0& {a}^{2}+{b}^{2}=0\\ \sqrt{\frac{{a}^{2}+{b}^{2}}{{a}^{2}}}& \mathrm{otherwise}\end{array}\right\\right)$
 $\sqrt{\frac{{{a}}^{{2}}{+}{{b}}^{{2}}}{{{a}}^{{2}}}}$ (7.2.3)
 >
 ${0}$ (7.2.4)

Piecewise conditions involving floor, ceil, round, frac, trunc can now be simplified:

 > $\mathrm{simplify}({\begin{array}{cc}f\left(x\right)& \mathrm{round}\left(x\right)>\frac{1}{2}\\ g\left(x\right)& \mathrm{otherwise}\end{array})$
 $\left\{\begin{array}{cc}{g}{}\left({x}\right)& {x}{<}\frac{{1}}{{2}}\\ {f}{}\left({x}\right)& \frac{{1}}{{2}}{\le }{x}\end{array}\right\$ (7.2.5)

Branch conditions other than equations, inequations, and inequalities are now taken into account while simplifying branch values:

 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}⌊x⌋& x\colon\colon \mathrm{integer}\\ {x}^{2}& \mathrm{otherwise}\end{array}\right\\right)$
 $\left\{\begin{array}{cc}{x}& {x}{::}{'}{\mathrm{integer}}{'}\\ {{x}}^{{2}}& {\mathrm{otherwise}}\end{array}\right\$ (7.2.6)
 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}{\left({x}^{3}\right)}^{1}{3}}& x\colon\colon \mathrm{posint}\\ x& \mathrm{otherwise}\end{array}\right\\right)$
 ${x}$ (7.2.7)

Branch conditions are now simplified more effectively using basic boolean logic:

 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}0& a=1\vee \left(a=1\wedge b\colon\colon \mathrm{integer}\right)\\ 1& \mathrm{otherwise}\end{array}\right\\right)$
 $\left\{\begin{array}{cc}{0}& {a}{=}{1}\\ {1}& {\mathrm{otherwise}}\end{array}\right\$ (7.2.8)
 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}f\left(x\right)& {x}^{2}={y}^{2}\vee {x}^{2}<{y}^{2}\\ g\left(x\right)& \mathrm{otherwise}\end{array}\right\\right)$
 $\left\{\begin{array}{cc}{f}{}\left({x}\right)& {{x}}^{{2}}{\le }{{y}}^{{2}}\\ {g}{}\left({x}\right)& {\mathrm{otherwise}}\end{array}\right\$ (7.2.9)

simplify now reorders piecewise conditions when appropriate:

 > $\mathrm{simplify}\left(\left(\left\{\begin{array}{cc}4& a=b\\ 5& a\ne b\end{array}\right\\right)-\left(\left\{\begin{array}{cc}5& a\ne b\\ 4& a=b\end{array}\right\\right)\right)$
 ${0}$ (7.2.10)

Piecewise conditions are now better normalized;

 >
 ${0}$ (7.2.11)

Common terms and factors are now pulled out of piecewise branch values where possible:

 > $\mathrm{simplify}\left(\left(\left\{\begin{array}{cc}y& z-y<\frac{1}{2}\\ y+1& \mathrm{otherwise}\end{array}\right\\right)-\left(\left\{\begin{array}{cc}2y& z-y<\frac{1}{2}\\ 2y+1& \mathrm{otherwise}\end{array}\right\\right)\right)$
 ${-}{y}$ (7.2.12)
 > $\mathrm{simplify}\left(\left\{\begin{array}{cc}-4\mathrm{\pi }{\mathrm{_C8}}_{\mathrm{_Z1},\mathrm{_Z2}}{\mathrm{BesselJ}\left(\mathrm{_Z1},\mathrm{BesselJZeros}\left(\mathrm{_Z1},\mathrm{_Z2}\right)r\right)}^{2}r& x=0\\ -2\mathrm{\pi }{\mathrm{_C8}}_{\mathrm{_Z1},\mathrm{_Z2}}{\mathrm{BesselJ}\left(\mathrm{_Z1},\mathrm{BesselJZeros}\left(\mathrm{_Z1},\mathrm{_Z2}\right)r\right)}^{2}r& \mathrm{otherwise}\end{array}\right\\right)$
 $\left(\left\{\begin{array}{cc}{-4}& {\mathrm{_Z1}}{=}{0}\\ {-2}& {\mathrm{otherwise}}\end{array}\right\\right){}{\mathrm{\pi }}{}{{\mathrm{_C8}}}_{{\mathrm{_Z1}}{,}{\mathrm{_Z2}}}{}{{\mathrm{BesselJ}}{}\left({\mathrm{_Z1}}{,}{\mathrm{BesselJZeros}}{}\left({\mathrm{_Z1}}{,}{\mathrm{_Z2}}\right){}{r}\right)}^{{2}}{}{r}$ (7.2.13)
 • Nonpiecewise-related improvements made to simplify:

Improved simplification of $\mathrm{Gamma}$:

 >
 ${0}$ (7.2.14)

Trig functions are now expanded if it helps with simplification:

 > $\mathrm{simplify}\left(\mathrm{sec}\left(z\right)+I\left(I\mathrm{tan}\left(\frac{\mathrm{\pi }}{4}+\frac{z}{2}\right)-I\mathrm{tan}\left(z\right)\right)\right)$
 ${0}$ (7.2.15)

Simplification of expressions containing arctan has been improved:

 > $\mathrm{simplify}\left(\mathrm{sin}\left(\frac{\mathrm{arctan}\left(\frac{\sqrt{111}}{9}\right)}{3}+\frac{\mathrm{\pi }}{6}\right)-\mathrm{cos}\left(\frac{\mathrm{arctan}\left(\frac{3\sqrt{111}}{5}\right)}{6}+\frac{\mathrm{\pi }}{6}\right),\mathrm{arctrig}\right)$
 ${0}$ (7.2.16)

Expressions containing csgn can now be more effectively simplified:

 > $\mathrm{simplify}\left(-\sqrt{{y}^{2}}\mathrm{csgn}\left(y\right)+y,\mathrm{csgn}\right)$
 ${0}$ (7.2.17)

Conversion between powers, exponentials, trig functions, and radicals to achieve simplification has been improved:

 > $\mathrm{simplify}\left(\mathrm{cos}\left(x\right)-\frac{{ⅇ}^{Ix}}{2}\right)$
 $\frac{{{ⅇ}}^{{-I}{}{x}}}{{2}}$ (7.2.18)
 > $\mathrm{simplify}\left(\mathrm{sinh}\left(x\right)-\mathrm{cosh}\left(x\right)+{ⅇ}^{-x}\right)$
 ${0}$ (7.2.19)
 > $\mathrm{simplify}\left(\left(-3\mathrm{sin}\left(x\right)+9I\right){ⅇ}^{-\frac{I}{3}\mathrm{sin}\left(x\right)}+\left(-9I-3\mathrm{sin}\left(x\right)\right){ⅇ}^{\frac{I}{3}\mathrm{sin}\left(x\right)}+6\mathrm{sin}\left(x\right)\mathrm{cos}\left(\frac{\mathrm{sin}\left(x\right)}{3}\right)-18\mathrm{sin}\left(\frac{\mathrm{sin}\left(x\right)}{3}\right)\right)$
 ${0}$ (7.2.20)
 > $\mathrm{simplify}\left(-{\left(z+1\right)}^{\frac{b}{2}+\frac{1}{2}}{ⅇ}^{\left(b+1\right)\mathrm{\pi }I}\left({z}^{-a-b-1}-{z}^{-a-b+1}\right){\left(z-1\right)}^{\frac{b}{2}+\frac{1}{2}}+{\left(z-1\right)}^{\frac{3}{2}+\frac{b}{2}}{\left(z+1\right)}^{\frac{3}{2}+\frac{b}{2}}{ⅇ}^{\left(b+2\right)\mathrm{\pi }I}{z}^{-a-b-1}\right)$
 ${0}$ (7.2.21)

Symbolic powers of integers are now combined more effectively:

 > $\mathrm{simplify}\left({\left(-1\right)}^{2k}\left({4}^{k}{25}^{k}-{100}^{k}\right)\right)$
 ${0}$ (7.2.22)

simplify now rewrites expressions using a common integer base:

 > $\mathrm{simplify}\left({\left(-1\right)}^{\mathrm{_k2}}{4}^{\mathrm{_k2}}{2}^{-\mathrm{\nu }}{16}^{-\mathrm{_k2}}\right)$
 ${{2}}^{{-}{2}{}{\mathrm{_k2}}{-}{\mathrm{\nu }}}{}{\left({-1}\right)}^{{\mathrm{_k2}}}$ (7.2.23)

Radicals are now typically combined by simplify:

 > $\mathrm{simplify}\left(\sqrt{3}\sqrt{2}\right)$
 $\sqrt{{6}}$ (7.2.24)

If appropriate conditions are satisfied, certain simplifications of floor, ceil, and round are applied:



 >
 $⌊{2}{}{z}⌋$ (7.2.25)
 > $\mathrm{simplify}\left(⌊z⌉-⌊2z⌋\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(z-\frac{1}{2}\right)::\left(\mathrm{real}\wedge ¬\mathrm{negint}\right)$
 ${-}⌊{z}⌋$ (7.2.26)

now simplifies:

 >
 ${\left({-1}\right)}^{{n}}{}{\mathrm{EllipticE}}{}\left({x}\right)$ (7.2.27)

solve

Description

 • The solve command in Maple 2019 has undergone several improvements.

solve Examples

Maple2019 solves equations with inequalities more carefully:

 > $\mathrm{solve}\left(\sqrt{x}\le 0\right)$
 ${0}$ (8.2.1)
 > $\mathrm{solve}\left(-\frac{1}{\sqrt{x}}\le 2\right)$
 $\left({0}{,}{\mathrm{\infty }}\right)$ (8.2.2)
 >
 $\left[\left\{{x}{=}{0}{,}{y}{=}{0}\right\}\right]$ (8.2.3)
 >
 $\left\{{x}{<}{-6}\right\}{,}\left\{{-5}{<}{x}{,}{x}{<}{0}\right\}$ (8.2.4)

Other Improvements

Description

 • There are other commands which have improved.

Other examples

minimize can now solve this example:

 >
 ${0}$ (9.2.1)

expand now takes into account more assumptions:

 >
 ${0}$ (9.2.2)

floor and ceil now make better use of assumptions:

 >
 ${0}$ (9.2.3)

rationalize works better on certain examples of nested radicals:

 >
 $\frac{{\left(\frac{{y}}{{{x}}^{{3}}{{2}}}}\right)}^{{2}}{{3}}}{}{{x}}^{{9}}{{2}}}}{{{y}}^{{3}}}$ (9.2.4)

Expressions with nested calls to Re and Im now evaluate better:

 >
 ${{\mathrm{\Re }}{}\left({x}\right)}^{{2}}{+}{\mathrm{\Im }}{}\left({{x}}^{{2}}\right)$ (9.2.5)