The PhD thesis by Paul Wang, at MIT, describes the initial approach to doing
definite integrals by contour integration in Macsyma. (written c. 1972).
The methods used in Mathematica may be a superset of this, and may be
described somewhere.
The reason definite integrals are NOT done first by plugging in to the
indefinite integral is simply this:
If you did that first, you might get an answer, but what would you do with
it? It might be wrong!
Example:
Integrate(1/x^2,x,-1,1),
If you look at the indefinite integral, -1/x, and plug in upper/lower
values, you get an answer of -2.
If you look at 1/x^2, you might notice that for any real x, it is positive.
So the integral between -1 and 1 cannot be negative.
RJF
> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu] On Behalf Of Jay Belanger
> Sent: Monday, April 16, 2007 9:21 AM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] integration asks to many questions
>
>
> Raymond Toy <toy.raymond at gmail.com> writes:
> ...
> >> Maxima's algorithms for definite integrals are mostly different
> >> from the algorithms for indefinite integrals. Maxima might try
> >> to evaluate a definite integral via the fundamental theorem of
> >> calculus but usually it tries other methods first (if I
> recall correctly).
> >>
> >>
> > Yes, that's exactly right. Maxima tries to convert
> definite integrals
> > to contour integrals which can be evaluated using residues.
>
> Really? That's pretty neat.
> I've only come across residues for improper integrals or very special
> forms. How does this work more generally?
>
> Jay
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