Advanced Math

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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