Complex analysis - conformal maps -mapping

[sweetvirgogirl]

find a one-to-one analytic function that maps the domain {} to upper half plane etc ...

for questions like these, do we just have to be blessed with good intuition or there are actually sound mathematical ways to come up with one-to-one analytic functions that satisfy the given requirement?

for example, for the domain {z:|z| > R and I am z > 0} ... if we have to find a one-to-one analytic function that maps the domain to the upper half plane... how do we do that? i know the final answer ... but i need mathematical reasoning ... i need someone to tell me step by step what's going on ...
the book doesn't explain this section well ...

 

[matt grime]

There are general methods, though I can'te recall them exactly, but for the cases you're most likely to meet remember translation, angle expansion, and how to map strips to the upper half plane. Putting those together will sort out the easiest cases. And remember to work on where the boundary goes.

 

[sweetvirgogirl]

There are general methods, though I can'te recall them exactly, but for the cases you're most likely to meet remember translation, angle expansion, and how to map strips to the upper half plane. Putting those together will sort out the easiest cases. And remember to work on where the boundary goes.

can someone here go over " translation, angle expansion, and how to map strips to the upper half plane. "?
and can someone go over how to put them together?

lol you see my problem? i have no clue how to even have a start

 

[matt grime]

say you wanted to map the set of points {z : re(z)>1, Im(z)>0} to the upper half plane. Translate be subtracting one to get the upper right quadrant, then square to get the upper half plane... I'm trying to push you in the direction of things that are in your notes.

 

[mathwonk]

just recall the geometric meaning of complex addition and multiplication. adding is translation, and raising to the nth power is multiplyinbg the angle by n and also raising the length to the nth power.

so squaring for instance, maps the upper right quartile to the upper half plane, because every angle, measured from the positive real axis, gets doubled.


after these basic facts, then study the geometry of multiplicative inversion, which takes the poutside of the unit circle to the inside, except for the origin.

then complex conjugation, takes the upper half plane to the lower one.

then notice what division does, of the sort |z-a|/|z-b|. i.e. |z-a| is the distance from z to a, so |z-a|/|z-b| < 1, says that z is closer to a than to b, which says by euclidean geometry, that z lies above the line where points are equidistant from a and b, i.e. z lies above the perpendicular bisector of the line joining a and b.

so |z-i|/|z+i| < 1, says z is in the upper half plane. thus w = (z-i)/(z+i) says that |w| < 1 iff z is in the upper half plane. so this maps the upper half plane into the unit disc.

just play with these relations a while.

 

[mathwonk]

here's a cute problem that a friend of mine and i disagree on the answer to:

suppose U is a simply connected open set in the complex plane and f:U-->V is a complex analytic function with simply connected image set V = f(U).

if the derivative of f is never zero in U, must f be injective?(i say no.)

 

[sweetvirgogirl]

what effect does multiplying by i have?

i am still lost ...
can you direct me to a list of those "meanings of geometric addition" etc ... maybe i am taking more complicated approach than i should be ... but things don't make much sense

 

[shmoe]

muliplying by i is a rotation by pi/2 radians counterclockwise about the origin. Think about the polar form of a complex number is the easiest to visualize the the geometric effect of multiplication. Addition looks the same as vector addition in R^2.