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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), y(x))](AdvancedMath/AdvancedMath_1.gif) |
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, `*`(x, `*`(BesselI(2, x)))), y(x))](AdvancedMath/AdvancedMath_3.gif) |
int
Description
- There have been various improvements made to the int command for Maple 2019.
int Examples
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![`assuming`([int(abs(`+`(cos(`*`(`+`(p, `-`(1)), `*`(x))), `-`(cos(`*`(`+`(p, 1), `*`(x)))))), x = 0 .. Pi)], [`>`(p, 1)])](AdvancedMath/AdvancedMath_13.gif) |
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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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](AdvancedMath/AdvancedMath_27.gif) |
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](AdvancedMath/AdvancedMath_31.gif) |
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](AdvancedMath/AdvancedMath_35.gif) |
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](AdvancedMath/AdvancedMath_39.gif) |
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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![`assuming`([simplify(piecewise(`<`(m, 2), floor(`+`(`*`(`/`(1, 2), `*`(m)))), 0))], [m::posint])](AdvancedMath/AdvancedMath_47.gif) |
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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![simplify(piecewise(x = 0, `+`(`-`(`*`(4, `*`(Pi, `*`(_C8[_Z1, _Z2], `*`(`^`(BesselJ(_Z1, `*`(BesselJZeros(_Z1, _Z2), `*`(r))), 2), `*`(r))))))), `+`(`-`(`*`(2, `*`(Pi, `*`(_C8[_Z1, _Z2], `*`(`^`(Besse...](AdvancedMath/AdvancedMath_65.gif) |
- Nonpiecewise-related improvements made to simplify:
Improved simplification of 
:
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![`assuming`([simplify(`/`(1, `*`(GAMMA(`+`(1, `-`(n))), `*`(GAMMA(n)))))], [n::integer])](AdvancedMath/AdvancedMath_68.gif) |
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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![`assuming`([simplify(`+`(floor(`+`(z, `/`(1, 2))), floor(z)))], [z::real])](AdvancedMath/AdvancedMath_90.gif) |
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![`assuming`([simplify(`+`(round(z), `-`(floor(`+`(`*`(2, `*`(z)))))))], [(`+`(z, `-`(`/`(1, 2))))::(And(real, Not(negint)))])](AdvancedMath/AdvancedMath_92.gif) |
now simplifies:
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![`assuming`([simplify(EllipticE(`^`(-1, n), x))], [n::integer])](AdvancedMath/AdvancedMath_95.gif) |
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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![[solve(`<=`(sqrt(x), `+`(`-`(`*`(`^`(y, 2))))), {x, y})]; 1](AdvancedMath/AdvancedMath_101.gif) |
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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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![`assuming`([floor(w)], [`>=`(w, 0), `<`(w, `/`(1, 2))])](AdvancedMath/AdvancedMath_109.gif) |
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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![Typesetting:-mprintslash([`+`(y(x), `*`(2, `*`(diff(y(x), x))), `*`(4, `*`(x, `*`(diff(y(x), `$`(x, 2))))))], [`+`(y(x), `*`(2, `*`(diff(y(x), x))), `*`(4, `*`(x, `*`(diff(diff(y(x), x), x)))))])](AdvancedMath/AdvancedMath_2.gif){kind=small}
![Typesetting:-mprintslash([`+`(`*`(`/`(1, 2), `*`(PIECEWISE([`+`(`-`(`*`(`^`(ceil(`+`(x, `-`(`/`(1, 2)))), 2))), `*`(x, `*`(PIECEWISE([`+`(`*`(2, `*`(x)), `-`(1)), (`+`(x, `/`(1, 2)))::integer], [`+`(`...](AdvancedMath/AdvancedMath_20.gif)
![Typesetting:-mprintslash([PIECEWISE([`+`(`-`(`*`(`+`(I), `*`(Pi))), Ei(`+`(ln(`+`(`-`(x))), `*`(I, `*`(Pi))))), `<`(x, 0)], [0, x = 0], [Ei(ln(x)), `<`(x, 1)], [`+`(`-`(infinity)), x = 1], [Ei(ln(x)),...](AdvancedMath/AdvancedMath_24.gif)
![Typesetting:-mprintslash([ex1 := `/`(`*`(exp(`+`(`-`(`*`(`/`(1, 2), `*`(t))))), `*`(sin(t))), `*`(`+`(1, exp(`+`(`-`(t)))), `*`(t)))], [`/`(`*`(exp(`+`(`-`(`*`(`/`(1, 2), `*`(t))))), `*`(sin(t))), `*`...](AdvancedMath/AdvancedMath_26.gif)
![Typesetting:-mprintslash([ex2 := `*`(exp(`+`(`*`(`/`(1, 2), `*`(t)))), `*`(ln(`+`(1, exp(`+`(`-`(t))))), `*`(cos(t))))], [`*`(exp(`+`(`*`(`/`(1, 2), `*`(t)))), `*`(ln(`+`(1, exp(`+`(`-`(t))))), `*`(co...](AdvancedMath/AdvancedMath_30.gif){kind=small}
![Typesetting:-mprintslash([ex3 := `/`(`*`(`+`(`*`(exp(`+`(`-`(t))), `*`(Heaviside(t))), `*`(exp(t), `*`(Heaviside(`+`(`-`(t))))))), `*`(`+`(`*`(2, `*`(t)), `*`(2, `*`(I)))))], [`/`(`*`(`+`(`*`(exp(`+`(...](AdvancedMath/AdvancedMath_34.gif){kind=small}
![Typesetting:-mprintslash([ex4 := `/`(`*`(csch(t), `*`(sin(t))), `*`(t))], [`/`(`*`(csch(t), `*`(sin(t))), `*`(t))])](AdvancedMath/AdvancedMath_38.gif){kind=small}
![Typesetting:-mprintslash([PIECEWISE([5, Or(x = 1, x = 2)], [6, otherwise])], [piecewise(Or(x = 1, x = 2), 5, 6)])](AdvancedMath/AdvancedMath_42.gif)
![Typesetting:-mprintslash([PIECEWISE([cos(T), (`+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(2, `*`(T))), Pi))), `*`(Pi)))))::integer], [1, otherwise])], [piecewise((`+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`...](AdvancedMath/AdvancedMath_44.gif)
![Typesetting:-mprintslash([PIECEWISE([g(x), `<`(x, `/`(1, 2))], [f(x), `<=`(`/`(1, 2), x)])], [piecewise(`<`(x, `/`(1, 2)), g(x), `<=`(`/`(1, 2), x), f(x))])](AdvancedMath/AdvancedMath_50.gif)
![Typesetting:-mprintslash([PIECEWISE([x, x::integer], [`*`(`^`(x, 2)), otherwise])], [piecewise(x::integer, x, `*`(`^`(x, 2)))])](AdvancedMath/AdvancedMath_52.gif)
![Typesetting:-mprintslash([PIECEWISE([0, a = 1], [1, otherwise])], [piecewise(a = 1, 0, 1)])](AdvancedMath/AdvancedMath_56.gif)
![Typesetting:-mprintslash([PIECEWISE([f(x), `<=`(`*`(`^`(x, 2)), `*`(`^`(y, 2)))], [g(x), otherwise])], [piecewise(`<=`(`*`(`^`(x, 2)), `*`(`^`(y, 2))), f(x), g(x))])](AdvancedMath/AdvancedMath_58.gif)
![Typesetting:-mprintslash([`*`(PIECEWISE([-4, _Z1 = 0], [-2, otherwise]), `*`(Pi, `*`(_C8[_Z1, _Z2], `*`(`^`(BesselJ(_Z1, `*`(BesselJZeros(_Z1, _Z2), `*`(r))), 2), `*`(r)))))], [`*`(piecewise(_Z1 = 0, ...](AdvancedMath/AdvancedMath_66.gif)
![[{x = 0, y = 0}]](AdvancedMath/AdvancedMath_102.gif){kind=small}
