Bill Page wrote:
>
> Perhaps you recall the attached email from a few years ago where you
> discussed multivariate Taylor series? Recently I wanted to do
> something in FriCAS similar to what is done easily in Maple, such as:
>
> mtaylor(sinh(x)*cosh(y),[x,y])
>
> x + 1/2 x y^2 + 1/6 x^3 + 1/24 x y^4 + ...
>
> If I try this in FriCAS
>
> x:=taylor 'x
> y:=taylor 'y
> sinh(x)*cosh(y)
>
> what I get is not a multivariate Taylor series. But if I do this, for example
>
> sinh(x*y)
>
> then the result does look like a multivariate series but the type is
>
> UnivariateTaylorSeries(UnivariateTaylorSeries(Expression(Integer),x,0), y,0)
>
> so this is not quite what I expected either.
>
> Am I missing something important about how to do this or is there
> really something missing in the way FriCAS does this?
ATM FriCAS has no multivariate series expander. In cases like
above one could manufacture an expander by iterating univariate
expansions. However, this is likely to be inefficient and
I am not sure if it always work.
Basically, the problem is that expression may contain
apparent singularities which cancel. Series expander
is supposed to get result after cancelation of
singularities. I think I know what power series
expander is doiung, but this is somewhat ugly and
I still need some time to convince myself that
this is correct. To get idea of the problem look
at expansion of f(x) = sin(x+y)/sin(x*y) produced by
Maple. Note: the function above mathematically has
no Taylor expansion, but g(x) tan(x*y)/sin(x*y) has.
Our expander handling g effectively has to produce
expansion of tan(x8y) and 1/sin(x*y), so in intermediate
stages we need to go beyond Taylor series.
--
Waldek Hebisch
[email protected]
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