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.

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**int**

Description

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

int Examples

- New results from int:

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- Improved answers for definite integrals when the AllSolutions option is given:

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**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:

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**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.

**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;

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

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

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

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simplify now reorders piecewise conditions when appropriate:

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Piecewise conditions are now better normalized;

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

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- Nonpiecewise-related improvements made to simplify:

Improved simplification of :

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Trig functions are now expanded if it helps with simplification:

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Simplification of expressions containing arctan has been improved:

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Expressions containing csgn can now be more effectively simplified:

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

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Symbolic powers of integers are now combined more effectively:

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simplify now rewrites expressions using a common integer base:

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Radicals are now typically combined by simplify:

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

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now simplifies:

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**solve **

Description

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

solve Examples

Maple2019 solves equations with inequalities more carefully:

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**Other Improvements**

Description

- There are other commands which have improved.

Other examples

minimize can now solve this example:

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expand now takes into account more assumptions:

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floor and ceil now make better use of assumptions:

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rationalize works better on certain examples of nested radicals:

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

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