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Advanced Math

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

 

Differential Equations

int

Integral Transforms

Multivariate Limits

Real Roots of Polynomials

Residue

simplify

solve

Other Improvements

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.

DEtoolsFindODEcossqrtx,yx

yx+2ⅆⅆxyx+4xⅆ2ⅆx2yx

(1.1)

DEtoolsFindODEBesselI0,x + x BesselI2,x, yx

x42x34x23xyx+x3+x+4ⅆⅆxyx+x4+2x3+x2+2xⅆ2ⅆx2yx

(1.2)

int

Description

• 

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

int Examples

• 

New results from int:

0πsinx2ⅆx

FresnelS2π2π2+FresnelS22πFresnelS22π+FresnelS322π

(2.2.1)

0πⅇIsinxⅆx

πBesselJ0,1+IπStruveH0,1

(2.2.2)

0π2secu2ⅇsecuⅆu

BesselK1,1

(2.2.3)

01signumarccos1zⅆz

I

(2.2.4)

0πcosp1xcosp+1xⅆxassuming p>1

2sinπpsignumsinπpcosπpsinπpsignumsinπpsinπp+1pp+sinπpp+psinπppcosπpp2p2cosπp+1

(2.2.5)

0π2cscx1νcosxⅆx

πνsinνπ

(2.2.6)

02x212+Ilnⅇ2Iπx22πⅆx

523

(2.2.7)
• 

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

introundt,t=0..x,AllSolutions # Similar improvements with ceil, floor, frac, trunc instead of round.

x122+x2x1x+12::&apos;integer&apos;2xotherwise0<xx+122+x2x+1x12::&apos;integer&apos;2xotherwisex02

(2.2.8)

int12&plus;cost&comma;t&equals;0..x&comma;AllSolutions

2ππx2π+ππx2π::&apos;integer&apos;2arctantanx233otherwise0<x2ππx2π+2πππx2π::&apos;integer&apos;2arctantanx233otherwisex033

(2.2.9)

int1lnt&comma;t&equals;0..x&comma;AllSolutions&comma;CauchyPrincipalValue

Iπ+Eilnx+Iπx<00x=0Eilnxx<1x=1Eilnx1<x

(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:

ex1&ExponentialE;t21+&ExponentialE;tsintt

ex1&ExponentialE;t2sint1+&ExponentialE;tt

(3.2.1)

inttransfourierex1&comma;t&comma;s

I2lncscIπs12+Iπ+cotIπs12+IπcscIπs+12+Iπ+cotIπs+12+Iπ

(3.2.2)

ex2&ExponentialE;t2ln1+&ExponentialE;tcost

ex2&ExponentialE;t2ln1+&ExponentialE;tcost

(3.2.3)

inttransfourierex2&comma;t&comma;s

πcscπ12+Is12Is12I+cscπ12+Is+12Is1+2I

(3.2.4)

ex3&ExponentialE;tHeavisidet+&ExponentialE;tHeavisidet2t+I

ex3&ExponentialE;tHeavisidet+&ExponentialE;tHeavisidet2t+2I

(3.2.5)

inttransfourierex3&comma;t&comma;s

&ExponentialE;sEi1Is&ExponentialE;−I+Ei1Is&ExponentialE;I2

(3.2.6)

ex4cschtsintt

ex4cschtsintt

(3.2.7)

inttransfourierex4&comma;t&comma;s

−I2π+ln&ExponentialE;πs+1+1&ExponentialE;πs1+1

(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;

simplify5x=15x=26otherwise

5x=1x=26otherwise

(7.2.1)

simplify0π2=TcosT2Tπ2πinteger1otherwise

cosT2T+π2π::&apos;integer&apos;1otherwise

(7.2.2)

simplify0a2+b2=0a2+b2a2otherwise

a2+b2a2

(7.2.3)

simplifym2m < 20otherwiseassuming mposint

0

(7.2.4)

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

simplify&lpar;&lcub;fxroundx&gt;12gxotherwise&rpar;

gxx<12fx12x

(7.2.5)

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

simplifyxxintegerx2otherwise

xx::&apos;integer&apos;x2otherwise

(7.2.6)

simplifyx313xposintxotherwise

x

(7.2.7)

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

simplify0a=1a=1binteger1otherwise

0a=11otherwise

(7.2.8)

simplifyfxx2=y2x2<y2gxotherwise

fxx2y2gxotherwise

(7.2.9)

simplify now reorders piecewise conditions when appropriate:

simplify4a=b5ab5ab4a=b

0

(7.2.10)

Piecewise conditions are now better normalized;

simplifyf&lpar;x&rpar;xt20otherwisef&lpar;x&rpar;2 xt00otherwise

0

(7.2.11)

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

simplifyyzy<12y+1otherwise2yzy<122y+1otherwise

y

(7.2.12)

simplify4π_C8_Z1,_Z2BesselJ_Z1&comma;BesselJZeros_Z1&comma;_Z2r2rx=02π_C8_Z1,_Z2BesselJ_Z1&comma;BesselJZeros_Z1&comma;_Z2r2rotherwise

−4_Z1=0−2otherwiseπ_C8_Z1,_Z2BesselJ_Z1&comma;BesselJZeros_Z1&comma;_Z2r2r

(7.2.13)
• 

Nonpiecewise-related improvements made to simplify:

Improved simplification of Gamma:

simplify1Γ1nΓn assuming ninteger

0

(7.2.14)

Trig functions are now expanded if it helps with simplification:

simplifysecz+IItanπ4+z2Itanz

0

(7.2.15)

Simplification of expressions containing arctan has been improved:

simplifysinarctan11193+π6cosarctan311156+π6&comma;arctrig

0

(7.2.16)

Expressions containing csgn can now be more effectively simplified:

simplifyy2csgny+y&comma;csgn

0

(7.2.17)

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

simplifycosx&ExponentialE;Ix2

&ExponentialE;−Ix2

(7.2.18)

simplifysinhxcoshx+&ExponentialE;x

0

(7.2.19)

simplify3sinx+9I&ExponentialE;I3sinx+9I3sinx&ExponentialE;I3sinx+6sinxcossinx318sinsinx3

0

(7.2.20)

simplifyz+1b2+12&ExponentialE;b+1πIzab1zab+1z1b2+12+z132+b2z+132+b2&ExponentialE;b+2πIzab1

0

(7.2.21)

Symbolic powers of integers are now combined more effectively:

simplify−12k4k25k100k

0

(7.2.22)

simplify now rewrites expressions using a common integer base:

simplify−1_k24_k22ν16_k2

22_k2ν−1_k2

(7.2.23)

Radicals are now typically combined by simplify:

simplify32

6

(7.2.24)

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

simplifyz+12+zassuming zreal

2z

(7.2.25)

simplifyz2zassumingz12::real¬negint

z

(7.2.26)

EllipticE1n&comma;x now simplifies:

simplifyEllipticE−1n&comma;xassuming ninteger

−1nEllipticEx

(7.2.27)

solve

Description

• 

The solve command in Maple 2019 has undergone several improvements.

solve Examples

Maple2019 solves equations with inequalities more carefully:

solvex0

0

(8.2.1)

solve1x2

0&comma;

(8.2.2)

solvexy2&comma;x&comma;y &semi;

x=0&comma;y=0

(8.2.3)

solvexx+5x+6<0&comma;x&semi;  

x<−6,−5<x&comma;x<0

(8.2.4)

Other Improvements

Description

• 

There are other commands which have improved.

Other examples

minimize can now solve this example:

minimizex4+y4+z4x2+y2+z2&comma;x=−1..1&comma;y=−1..1&comma;z=−1..1&semi;  

0

(9.2.1)

expand now takes into account more assumptions:

expandcc1mc1cmcmc1cmcm+c1mcmassumingm&apos;integer&apos;&semi; 

0

(9.2.2)

floor and ceil now make better use of assumptions:

wassumingw0,w<12 

0

(9.2.3)

rationalize works better on certain examples of nested radicals:

rationalize1yx3273&semi; 

yx3223x92y3

(9.2.4)

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

x2+x2&semi; 

x2+x2

(9.2.5)