A few complex analysis questions


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?

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)?

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.

For example if we have to integrate something like (not necessarily
exactly this) 1/(a+x^(3/2)) from -oo to oo and then we use some kind
of keyhole contour(?). Why does this work. Can we somehow choose where
the branch cut is? I think I have read somewhere that we can choose
branch cut to be for example positive real axis (vs. negative real
axis). What does all of this mean and why does it work?

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?

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?

5. Winding numbers: How do I in general prove by using winding numbers
that a given map is one-to-one?

I think this is enough for now. I probably will be asking more
questions in the future. :)

.