@end example
Multivariate polynomials and rational functions may be expanded,
-collected and normalized (i.e. converted to a ratio of two coprime
-polynomials):
+collected, factorized, and normalized (i.e. converted to a ratio of
+two coprime polynomials):
@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
4*x*y-y^2+x^2
> expand(a*b);
8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
+> factor(%);
+(4*x*y+x^2-y^2)^2*(x^2+3*y^2)
> collect(a+b,x);
4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
> collect(a+b,y);
3*y^2+x^2
@end example
+Here we have made use of the @command{ginsh}-command @code{%} to pop the
+previously evaluated element from @command{ginsh}'s internal stack.
+
You can differentiate functions and expand them as Taylor or Laurent
series in a very natural syntax (the second argument of @code{series} is
a relation defining the evaluation point, the third specifies the
-Euler-1/12+Order((x-1/2*Pi)^3)
@end example
-Here we have made use of the @command{ginsh}-command @code{%} to pop the
-previously evaluated element from @command{ginsh}'s internal stack.
-
Often, functions don't have roots in closed form. Nevertheless, it's
quite easy to compute a solution numerically, to arbitrary precision:
The multiple polylogarithm is the most generic member of a family of functions,
to which others like the harmonic polylogarithm, Nielsen's generalized
polylogarithm and the multiple zeta value belong.
-Everyone of these functions can also be written as a multiple polylogarithm with specific
+Each of these functions can also be written as a multiple polylogarithm with specific
parameters. This whole family of functions is therefore often referred to simply as
multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While