Re: question and help please

On 31-03-2006 15:31, David C. Ullrich wrote:

if f : G -------> C is analytic except for poles , show that the set of
poles of f cannot have a limit point in G ..
Let z_0 be a point of G. Then either _f_ is defined at z_0 or z_0 is a
pole of _f_. If _f_ is defined at z_0, then there's a power series
sum_n a_n(z - z_0)^n which converges to f(z) in some disk centered at
z_0. So, that disk has no poles of _f_. If z_0 is a pole of _f_, then
there's some natural number _m_ and some power series
sum_n a_n(z - z_0)^n such that, for some r > 0, the disk D(z_0,r) is
contained in G and that, for each z in D(z_0,r)\{z_0},

f(z) = (sum_n a_n(z - z_0)^n)/(z - z_0)^m.

So, z_0 is the only pole of _f_ in D(z_0,r). This proves that there's
some neighborhood of z_0 such that it either has no poles or that z_0
itself is the only pole there.

Actually this doesn't really require proof. A pole is _by definition_
a certain sort of _isolated singularity_, so by definition if z_0
is a pole then there exists a neighborhood of z_0 in which there
are no other poles.

That's the definition that I use too, but I assumed that the OP's
definition of pole of a function _f_ was a point z_0 such that, for some
natural number _m_ and some power series sum_n a_n(z - z_0)^n,

f(z) = (sum_n a_n(z - z_0)^n)/(z - z_0)^m

near z_0. Of course, this *implies* that poles are isolated.

Best regards,

Jose Carlos Santos
.

Relevant Pages

  • Re: OT-got any ideas?
    ... neighborhood could be next!" ... i hate the poles, the one directly behind my house has to ... consent thru to a hearing with their own employees (i think ... they also want a 10 metre higher fence (they already added ...
    (rec.crafts.textiles.quilting)
  • Re: question and help please
    ... poles of f cannot have a limit point in G .. ... there's some natural number _m_ and some power series ... some neighborhood of z_0 such that it either has no poles or that z_0 ... David C. Ullrich ...
    (sci.math)
  • Re: Analytic continuation
    ... Murray Arnow wrote: ... singular points. ... Yes I agree but my question was really about the poles that were far enough away not to effect the power series in use - i.e. well outside the radius of convergence. ...
    (sci.physics.research)
  • Re: question and help please
    ... poles of f cannot have a limit point in G .. ... there's some natural number _m_ and some power series ... Best regards, ... Jose Carlos Santos ...
    (sci.math)