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Advanced Math

Maple 2018 includes numerous cutting-edge updates in a variety of branches of mathematics.

 

Integration

PolynomialTools

Simplification

Summation

Sturm Sequences

signum

is, coulditbe

Re, Im

max, min

arctan

SMTLIB

Integration

The int command has seen improvements to various methods. The following new results come from the asymptotic method:0sinxx3ⅆx

(1.1)

0cosxcosx3x2ⅆx

(1.2)

This new result results from an improvement in the elliptic integration code:

0101ⅇ2Iπx+ⅇ2Iπ yⅆxⅆy

4π

(1.3)

The next two examples are due to improvements to the Risch algorithm:

normalsinxarctanaa2+1x1+ⅇaⅆx

ⅇxlna2+1arctanatanxarctana22+2lna2+1tanxarctana2+arctanaⅇalna2+12+arctana2ⅇa+lna2+12+arctana21+tanxarctana22

(1.4)

83lnu23020162lnu2025+58497230375u52+4723lnu212603961877lnu297675+80288533431255875u72ⅆu

83u72lnu2105327088u72lnu99225+67139896u7210418625+4723u92lnu256708915584u92lnu2679075+5441402884u92843908625

(1.5)

This previously unsolved example is due to an improvement in the hyperexponential algorithm:
sinhx+coshxsinhx+1ⅆx

tanhx2+1tanhx21tanhx21arcsintanhx22+22+2+22242+2222+22arctanh1+tanhx22tanhx22+22tanhx22+12+22arcsintanhx2222+2222422+22+2arcsintanhx22+22+2arctantanhx22+tanhx212+22tanhx22+12arcsintanhx22+22+2+22242+22arcsintanhx2222+222242+22+2arctantanhx22+tanhx212+22tanhx22+12tanhx2+1tanhx212+22

(1.6)

PolynomialTools

Two new commands, Homogenize and IsHomogeneous, for performing and testing (weighted) homogenization were added to the PolynomialTools package.

withPolynomialTools:

fw5+w2x+w y+y3

fw5+w2x+y3+wy

(2.1)

IsHomogeneousf

false

(2.2)

gHomogenizef,z

gw5+w2xz2+wyz3+y3z2

(2.3)

IsHomogeneousg

true

(2.4)

Homogenizef,z,w,x,1,2

y3z5+wyz4+w5+w2xz

(2.5)

Simplification

The simplify command has been enhanced with respect to logarithms whose arguments are constant radical expressions. Some examples:

simplifyln12+ln1+2

Iπ

(3.1)

simplifyln2+2+2+2

ln2+Iπ8

(3.2)

simplify7 π ln23+11πln2+3+38ln31π+2πln1+3+4ln2π23ln22+3ln232+3ln2+32+43ln2ln31+43ln2ln1+343ln31243ln1+32

24πln2

(3.3)

simplify can now recognize more trig and exp simplifications:
simplifycos112Pi+sin112Pi+122

0

(3.4)

simplify16sin49Pi24sin19Pi32sin29Pi+83cos49Pi+83cos29Pi

0

(3.5)

simplifyⅇI5π54+14I2554

1

(3.6)

simplify now tries harder to simplify constants that appear as subexpressions:
simplify11/512/5+13/514/51x

0

(3.7)

Summation

The sum command with the parametric option has been improved for the case of hypergeometric sums with more than one parameter. The following sums used to return unevaluated in Maple 2017.

sumxnn+b, n = 0 .. infinity,parametric

FAILb=01bx=0b0FAIL1+b::'nonposint'LerchPhix,1,bx1x11+b::¬'nonposint'n=0xnn+botherwisex0b0

(4.1)

sumi+k1i zi, i = 0 .. infinity, parametric

11zkk::&apos;nonposint&apos;z=10<kz=1z1−1<kk0z<1i=0i+k1iziotherwise

(4.2)

sumk+azk&comma;k=0..&comma;parametric

az+a+zz12z<1k=0k+azkotherwise

(4.3)

Sturm Sequences

The commands sturm and sturmseq have been extended to support polynomials with real algebraic number coefficients.

fexpandx2x&plus;1x&plus;2 2

fx3+x22+x2+2x4x4

(5.1)

ssturmseqf&comma;x

sx3+x22+x2+2x4x4&comma;x2+22x3+2x3+2343&comma;x+252223112446&comma;1

(5.2)

sturms&comma;x&equals;..0

2

(5.3)

sturms&comma;x&equals;0..5

1

(5.4)

signum

The signum command has had various improvements made. The following examples previously returned with unevaluated signum calls:

signum~1x&comma;Ix assuming xreal

−1x&comma;Ix

(6.1)

signum~I2x&comma;I2x&plus;1 assuming xeven

1&comma;I

(6.2)

signum~I2x&comma;I2x&plus;1 assuming xodd

−1&comma;−I

(6.3)

signum6108&plus;12I6872&sol;3&plus;48108&plus;12I6871&sol;3108&plus;12I6872&sol;3&plus;126108&plus;12I6871&sol;3&plus;48108&plus;12I6872&sol;3&plus;48108&plus;12I6871&sol;3108&plus;12I6871&sol;3108&plus;12I6872&sol;3&plus;48108&plus;12I6871&sol;3

I

(6.4)

is, coulditbe

The is and coulditbe commands have had various improvements made. is performs more simplification than it did previously:

is&ExponentialE;&ExponentialE;1&comma;integer

true

(7.1)

isII2n&plus;1&gt;0 assuming neven

true

(7.2)

and takes into account more function properties:

is arccoshxIπ 0 assuming x<1

true

(7.3)

and operation properties:

iszn&comma;imaginaryassuming zimaginary&comma;ninteger&comma;znNonreal

true

(7.4)

The internal solver for systems of inequalities in is and coulditbe has been improved:

map2coulditbe&comma; p&comma; 1&comma; 1 assuming 1 < p&comma; 1 < q&comma; 1 < r&comma; 1 < p&plus;q&comma; 1 < p&plus;r&comma; 1 < r&plus;q&comma; 1 < p&plus;q&plus;r

false&comma;true

(7.5)

Re, Im

The Re and Im commands have had various improvements made. The following examples previously returned with unevaluated Re and Im calls:

z1 assuming z&gt;0&comma; z<1&comma; z1&equals;arccoshz

−Iz1

(8.1)

&comma;IΓ3212I3Γ32&plus;12I3

0&comma;Γ32I32Γ32+I32

(8.2)

max, min

In some cases, the max and min commands can now recognize numbers as real even though they are composed from nonreal ingredients:

max1&comma;124&plus;4I31&sol;3&plus;24&plus;4I31&sol;3

4+4I3132+24+4I313

(9.1)

evalc%

2cos2π9

(9.2)

evalf%&semi; 

1.532088886

(9.3)

arctan

The arctan command now performs some more automatic simplifications:

arctanb&plus;a2&plus;b2a assuming a &gt; 0&comma; b&gt;0

arctanab2+π2

(10.1)

arctant&plus;t2&plus;1 assuming t &gt; 0

arctan1t2+π2

(10.2)

arctan1&plus;z2&plus;1z

arctanz2

(10.3)

Furthermore, combine performs more simplifications on arctan functions:
combinearctan1z&plus;arctanz

csgnz2+1zπ2

(10.4)

combineexpandarctanbPi2arctanb&plus;b2&plus;1Pi assuming 0<b

12

(10.5)

 

SMTLIB

The SMTLIB package has been extended to support satisfiability queries on Boolean combinations of polynomial equations and inequalities.

Consider the following description of a set:

 

sys0y2+x2x33<2yy<x3+11<y0<x+y1x3yx<2y<2&colon;We can use first use SMTLIB[Satisfiable] to verify that a solution exists:
SMTLIBSatisfiablesys

true

(11.1)

 

In this simple two-dimensional case, we can use plots[inequal] to visualize the solution space:

withplots&colon;inequalsys&comma; x&equals;5..5&comma; y&equals;5..5

The new command SMTLIB[Satisfy] offers an efficient method of finding a concrete example for a point in the solution space:
SMTLIBSatisfysys

x=6&comma;y=217

(11.2)

To produce a satisfying point within the visual bounds of the plot above, we can simply augment our system with a bounding rectangle:
pt  SMTLIBSatisfysys and x&gt;5 and x<5 and y&gt;5 and y<5

x=−4&comma;y=−4

(11.3)

display inequalsys&comma; x&equals;5..5&comma; y&equals;5..5&comma; pointplot evalx&comma;y&comma;pt&comma; symbol&equals;solidcircle&comma; symbolsize&equals;25&comma; color&equals;red  &semi;