From: ikastan@alumni.caltech.edu (Ilias Kastanas) Newsgroups: sci.math Subject: Re: Convergence of series Date: 20 Sep 1997 19:35:31 GMT In article <3423EDC5.7C24@rug.ac.be>, Christophe Lermytte wrote: >David Ullrich wrote: >> >> Christophe Lermytte wrote: >> > >> > Can anyone tell me what is known about convergence >> > of a complex power series in points that are on the >> > circle of convergence ? >> >> Really a lot is known about this. To say >> "anything can happen" would be an exaggeration, but it's >> a good first approximation... Is there something >> specific you want to know about this? (Not that I'm >> going to know the answer, but in this case "what's >> known?" is simply too broad a question.) >> >> -- >> David Ullrich >> >> sig.txt not found > >I'm just very curious about the way the conv-points and div-points >are distributed on the circle. Is there anything you can say >about the location of these points ? Maybe (I'm just saying >something) the conv-points are always isolated, maybe they always form >a circle segment. Maybe (not likely, but it gives you an idea of what >I'm looking for) they are located in a halfplane Re z > a (cf. Laplace). > >Christophe Lermytte > >PS : * How can I do symbolic computations on this subject ? > I'm not an expert. I tried substituting z=rho*exp(i*theta) > and using the series for exp(z) but I got stuck. > * How can I do numerical computations on this subject ? > I've written a small QBASIC (or was it TurboPascal) program > that uses complex variables, used say 1000 instead of > infinity,tried to deal with overflows and so but in the > interesting regions there were to many oscillations to draw > conclusions. > * If many things are possible, maybe you can give me some examples Too many things are possible. E.g. the series doesn't "go" with the function. Take 1/(1+z) ; it is nice and regular at z=1, but its series, 1 - z + z^2 - ... diverges there. Now consider f(z) = Sum z^n/n^2; the series converges at z=1, but f satisfies df/dz = -log(1-z) /z and hence has a singularity at z=1. In fact if Sum a_n z^n has a_n >= 0 (and radius = 1) then z=1 has to be a singular point for the function. On the other hand, if a_n -> 0 then at least if f is regular at some z (|z| = 1) the series converges at such a z. A fun example is f(z) = 1 + z^2 + z^4 + z^8 + z^16 + ... The series diverges and the function is singular at z=1. The same is true for z with z^2 = 1 (note: f(z) = z^2 + f(z^2)). Likewise for z^4 = 1 (f(z) = z^2 + z^4 + f(z^4)). Same for z^8 = 1, z^16 = 1, ... So the singular points are dense on the unit circle. Again, a general theorem is hiding here. Ilias