conjugate and RootOf

The conjugate command now has extended support for RootOf expressions. You can now find the conjugate of the following:

 • Indexed RootOfs
 • RootOf expressions with a numerical selector and real coefficients

The following examples return unevaluated in Maple 2015 and earlier.

 > $\mathrm{conjugate}\left(\mathrm{RootOf}\left({x}^{5}+{x}^{2}-1,\mathrm{index}=2\right)\right)$
 ${\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{5}}{+}{{\mathrm{_Z}}}^{{2}}{-}{1}{,}{\mathrm{index}}{=}{5}\right)$ (1.1)
 >
 ${\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}\sqrt{{3}}{}{\mathrm{_Z}}{+}{1}{,}{-}{1}{-}{I}\right)$ (1.2)

eval

For indefinite integrals, sums, and products, as well as for differentiations, the eval command now supports an additive change of the variable.

 >
 ${∫}{f}{}\left({x}{+}{c}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (2.1)
 > $\genfrac{}{}{0}{}{\sum _{x}f\left(x\right)}{\phantom{x=a}}|\genfrac{}{}{0}{}{\phantom{\mathrm{f\left(x\right)}}}{x=x+c}$
 ${\sum }_{{x}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{f}{}\left({x}{+}{c}\right)$ (2.2)
 > $\genfrac{}{}{0}{}{f'\left(x\right)}{\phantom{x=a}}|\genfrac{}{}{0}{}{\phantom{\mathrm{f\left(x\right)}}}{x=x+c}$
 ${\mathrm{D}}{}\left({f}\right){}\left({x}{+}{c}\right)$ (2.3)
 > $\genfrac{}{}{0}{}{\prod _{x}f\left(x\right)}{\phantom{x=a}}|\genfrac{}{}{0}{}{\phantom{\mathrm{f\left(x\right)}}}{x=x+c}$
 ${\prod }_{{x}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{f}{}\left({x}{+}{c}\right)$ (2.4)

Gröbner Bases

Maple 2016 includes a new C implementation of the F4 algorithm for computing Gröbner bases, replacing FGb. The new code is generally faster and uses multiple threads. The benchmarks below show real time on a quad core Intel Core i5 4590 3.3 GHz computer using a logarithmic scale. The new code supports primes up to ${2}^{31}-1$, an increase over FGb's 16-bit primes. To compute over the rationals, Maple uses Chinese remaindering and rational reconstruction.

product

Maple 2016 includes improved handling of "product over RootOf" cases. The following example returns unevaluated in Maple 2015 and earlier:

 >
 ${{x}}^{{4}}{-}{2}$ (4.1)

Series and Limit Computations

A number of improvements were made to series and limit computations in Maple.

The following series and asymptotic functions were added:

 • Asymptotic expansions of Airy functions at $-\mathrm{∞}$
 • Series and asymptotic expansions of hypergeometric functions
 • Series expansions of abs and signum in the real case
 • Series expansion of GAMMA function at a symbolic pole
 • Asymptotic expansion of incomplete GAMMA function w.r.t. the parameter
 • Asymptotic expansion of Hurwitz Zeta function
 • Series and asymptotic expansions of harmonic numbers
 • Series expansions of ln and related functions with a logarithmic branch cut depending on a real parameter were improved.
 • Finally, limit computations of oscillating functions were improved.

New Series Expansions

The following series expansions could not be computed in earlier versions of Maple:

functions at 1

 > $\mathrm{series}\left(\mathrm{hypergeom}\left(\left[1,1\right],\left[-\frac{1}{3}\right],1-x\right),x\right)$
 ${-}\frac{{8}}{{9}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}}{{{x}}^{{7}{/}{3}}}{+}\frac{{32}}{{27}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}}{{{x}}^{{4}{/}{3}}}{-}\frac{{16}}{{81}}{}\frac{{\mathrm{π}}{}\sqrt{{3}}}{{{x}}^{{1}{/}{3}}}{+}\frac{{4}}{{7}}{-}\frac{{32}}{{729}}{}{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{2}{/}{3}}{+}\frac{{6}}{{35}}{}{x}{-}\frac{{40}}{{2187}}{}{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{5}{/}{3}}{+}\frac{{36}}{{455}}{}{{x}}^{{2}}{-}\frac{{64}}{{6561}}{}{\mathrm{π}}{}\sqrt{{3}}{}{{x}}^{{8}{/}{3}}{+}\frac{{81}}{{1820}}{}{{x}}^{{3}}{+}{\mathrm{O}}{}\left({{x}}^{{11}{/}{3}}\right)$ (5.1.1)

The case where the lower parameter minus the sum of the upper parameters is an integer is supported.

 > $\mathrm{series}\left(\mathrm{hypergeom}\left(\left[\frac{1}{2},\frac{3}{2}\right],\left[1\right],1-x\right),x\right)$
 $\frac{\frac{{2}}{{\mathrm{π}}}}{{x}}{+}\frac{{1}}{{2}}{}\frac{{-}{\mathrm{ln}}{}\left({x}\right){-}{1}{+}{4}{}{\mathrm{ln}}{}\left({2}\right)}{{\mathrm{π}}}{+}\frac{{1}}{{2}}{}\frac{{-}\frac{{3}}{{8}}{}{\mathrm{ln}}{}\left({x}\right){-}\frac{{13}}{{16}}{+}\frac{{3}}{{2}}{}{\mathrm{ln}}{}\left({2}\right)}{{\mathrm{π}}}{}{x}{+}\frac{{1}}{{2}}{}\frac{{-}\frac{{15}}{{64}}{}{\mathrm{ln}}{}\left({x}\right){-}\frac{{9}}{{16}}{+}\frac{{15}}{{16}}{}{\mathrm{ln}}{}\left({2}\right)}{{\mathrm{π}}}{}{{x}}^{{2}}{+}\frac{{1}}{{2}}{}\frac{{-}\frac{{175}}{{1024}}{}{\mathrm{ln}}{}\left({x}\right){-}\frac{{5255}}{{12288}}{+}\frac{{175}}{{256}}{}{\mathrm{ln}}{}\left({2}\right)}{{\mathrm{π}}}{}{{x}}^{{3}}{+}\frac{{1}}{{2}}{}\frac{{-}\frac{{2205}}{{16384}}{}{\mathrm{ln}}{}\left({x}\right){-}\frac{{11291}}{{32768}}{+}\frac{{2205}}{{4096}}{}{\mathrm{ln}}{}\left({2}\right)}{{\mathrm{π}}}{}{{x}}^{{4}}{+}{\mathrm{O}}\left({{x}}^{{5}}\right)$ (5.1.2)

The Γ function at a symbolic pole.

 >
 $\frac{\frac{{1}}{{\left({-}{1}\right)}^{{n}}{}{\mathrm{Γ}}{}\left({1}{-}{n}\right)}}{{x}{-}{n}}{+}\frac{{\mathrm{Ψ}}{}\left({1}{-}{n}\right)}{{\left({-}{1}\right)}^{{n}}{}{\mathrm{Γ}}{}\left({1}{-}{n}\right)}{+}\frac{\frac{{1}}{{6}}{}\frac{{{\mathrm{π}}}^{{2}}}{{\left({-}{1}\right)}^{{n}}}{-}\frac{\frac{{1}}{{2}}{}{\mathrm{Ψ}}{}\left({1}{,}{1}{-}{n}\right){}{\mathrm{Γ}}{}\left({1}{-}{n}\right){+}\frac{{1}}{{2}}{}{{\mathrm{Ψ}}{}\left({1}{-}{n}\right)}^{{2}}{}{\mathrm{Γ}}{}\left(\right)}{}}{}$