Hi Christophe,

On Sun, Feb 1, 2009 at 3:08 PM, Christophe Deroulers
<christophe.deroulers__ggsa...@normalesup.org> wrote:
>
>  Hello,
>
> Is there a way in Sage to express the derivative at one point of a
> "formal" function?

There's currently no way to do this right now.  I have a bit of code I
started at Sage Days 13 to allow one to do this.  I'll finish it up
and post it tonight.

For reference, there are two tickets for it at
http://trac.sagemath.org/sage_trac/ticket/3914 and
http://trac.sagemath.org/sage_trac/ticket/385.

> Actually, I would like to be able to show with Sage that f(x+a)+f(x-
> a)-2*f(x) is f''(x) at the first nonvanishing order in a, but it is
> still far away from working: Sage's "taylor" does not work with a
> "formal" function, and, even though Maxima does slightly better, it's
> not enough: for "taylor(f(x-a),a,0,2);", Maxima's result involves "d/
> da f(x-a) |a=0", while further simplification to -f'(x) would be
> needed.

You should ask on the Maxima list about getting Maxima to do such a
simplification natively.

--Mike

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