On Nov 15, 2007 2:49 AM, William Stein <[EMAIL PROTECTED]> wrote:
>
> On Nov 15, 2007 1:45 AM, Dan Drake <[EMAIL PROTECTED]> wrote:
> >
> > I discovered Sage recently and am very excited about it. In grad school,
> > I came to be very good with Mathematica, and am now doing a postdoc
> > where I only have access to (an old copy of) Maple. I like the idea of
> > having my mathematical software being only a 'wget' away, and even more
> > I like the idea of being able to see how it works and change it as I see
> > fit.
> >
> > But I'm having difficulty even defining a relatively simple function and
> > finding a floating-point approximation. I want to do
> >
> > f := x -> integrate(exp(t^2), t=0..x); (Maple)
> > evalf(f(2));
> >
> > f[x_] := Integrate[Exp[t^2], {t, 0, x}] (Mathematica)
> > N[f[2]]
> >
> > in Sage. I know Python, and Sage is built with Python, so I tried
> >
> > sage: def f(x): integrate(exp(t^2), t, 0, x)
> >
> > which complains that "global name 't' is not defined" when I try to
> > evaluate it. I don't want a global variable, just a local one to do the
> > integration!
>
> You can define that function and compute the integral in Sage as
> follows:
>
> sage: assume(x > 0)
> sage: f(x,t) = integrate(exp(t^2), t, 0, x)
> sage: f
> (x, t) |--> -sqrt(pi)*I*erf(I*x)/2
> sage: a = f(x,0)
> sage: a
> -sqrt(pi)*I*erf(I*x)/2
>
> The assume is needed above because of how Maxima works (you'll be
> non-interactively
> asked for it if you don't give it).
>
> Unfortunately, Sage does not have an implementation of computing
> a numerical approximation of erf(a) when a is not real, as PARI only
In that case, what is going on here?
sage: w = var("w")
sage: f = lambda w: exp((w+sqrt(-1))^2)
sage: f(w)
e^(w + I)^2
sage: f(w).nintegral(w,0,2)[0]
-4.4746871004750179
sage: f = lambda w: real(exp((w+sqrt(-1))^2))
sage: f(w).nintegral(w,0,2)[0]
-4.4746871004750179
sage: f = lambda w: imag(exp((w+sqrt(-1))^2))
sage: f(w).nintegral(w,0,2)[0]
-1.2224121101763701
> provides this function in case a is real, and maxima also seems to
> only provide it in that case. Thus you can't actually numerically evaluate
> the result at some point.
>
> You are the second person to request numerical evaluation of erf... at a
> complex argument this week (Paul Zimmerman requested the same
> thing a few days ago). So I hope one of the Sage developers looks into
> this. I've made it:
>
> http://trac.sagemath.org/sage_trac/ticket/1173
>
> I also made
>
> http://trac.sagemath.org/sage_trac/ticket/1174
>
> which is related.
>
> > What's the right way to go about all this? Like I said, I'm very excited
> > about Sage, but this is not such an auspicious start.
>
> Calculus is actually not at all the main focus of Sage or Sage development
> yet, unfortunately, though the calculus package is very powerful. It's
> something we introduced pretty recently into Sage, but it will still take
> some time to get every reasonable thing implemented, which very much
> is our goal. We need more people-power though, at least for the calculus
> part of Sage (where there are currently very few developers compared
> to other parts of Sage).
>
> -- William
>
>
> >
>
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