Re: A few complex analysis questions
- From: Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Sun, 11 Nov 2007 21:16:27 -0600
brutuz86@xxxxxxxxx writes:
Hi,
I hope you could help me understand better some complex analysis
topics.
1. Is there a some general nice way of constructing one-to-one
analytics maps from a given region onto another. I haven't studied all
of these things yet, but I have understood that by Riemann mapping
theorem you always can do it for simply connected open sets(?). How
about unbounded sets? I quess you can construct specific maps by using
for example linear fractional transformations. But do they always (or
almost always) work?
The Riemann mapping theorem shows that for any two simply connected open
subsets A and B of the complex plane, neither of which is the whole plane,
there is a one-to-one analytic mapping from A onto B. Unbounded sets are
included. In principle these mappings can be constructed, but it's not
necessarily simple or "closed-form".
For example: how would I map analyticly the region defined by Re-axis,
Im-Axis and the equation x^4-y^4=1 in the first quadrant one-to-one
onto the lower half plane? I think this should be possible. (So how do
I construct the spesific map and what is it)?
Probably not closed-form.
2. I am not completely comfortable with these branch points etc.
things and the log -function. What is this all about. If we do
integrals by method of residues why is it that we can use these
"branch things" to calculate the integrals.
If you want a single-valued holomorphic function, you need to remove
a branch cut from the domain. The contour is chosen to avoid the
branch cut. Inside the contour the function is nice and holomorphic,
so the Residue Theorem applies.
3. Related to 2. How do I show that a single-valued analytic branch of
sqrt(1-z^2) ca be defined in any region for which the points +-1 are
in the same component of the complement of the region?
The basic idea is this. Think of going around some loop in your region.
Since +-1 are in the same component of the complement, the number of times you
go around +1 is the same as the number of times you go around -1. As a
consequence, the change in argument of 1-z^2 = (1-z)(1+z) is a multiple
of 4 pi, and so the change in argument of sqrt(1-z^2) is a multiple of
2 pi: you come back with the same value of sqrt(1-z^2) as you started out with.
4. What is this thing with these poles and essential singularities
really? Basically, what is the real difference? How do I prove that an
isolated singularity of f(z) is removable as soon as either Re f(z) or
Im f(z) is bounded above or below?
The difference is exactly according to the definitions.
Prove it by showing this is neither a pole nor an essential singularity.
5. Winding numbers: How do I in general prove by using winding numbers
that a given map is one-to-one?
If f is analytic on a simply connected region G and is not one-to-one, say
f(a) = f(b), then if you take a simple closed contour C in G enclosing both,
with f(a) not on f(C), the winding number of f(C) around f(a) must be at least
2. So if you show that the winding number of f(C) around any point is
at most 1 (for a suitable family of contours C), ...
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
- References:
- A few complex analysis questions
- From: brutuz86
- A few complex analysis questions
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