# Homework Help: Composing functions that are conformal

1. Apr 8, 2012

### d2j2003

1. The problem statement, all variables and given/known data
let f and g be conformal analytic functions with the range of f a subset of the domain of g. Show that the composition g(f(z)) is also conformal.

2. Relevant equations

3. The attempt at a solution

Well I know that if the function is conformal it is 1-1, analytic, and its derivative is non zero. So i'm thinking that I would need to show that since f'(z)≠0 and g'(z)≠0 then (g(f(z))'≠0 and then show that since f and g are analytic, g(f(z)) is also analytic. Then the composition will be both analytic and its derivative will not be zero so it will be conformal. Am I working in the right direction?

2. Apr 8, 2012

### Office_Shredder

Staff Emeritus
That's one way to do it. The other way is to just go straight with the definition and show that g(f(z)) preserves angles

3. Apr 8, 2012

### d2j2003

Is it possible to show, in general, that g(f(z)) preserves angles? I thought we would have to have actual functions defined for that..

4. Apr 8, 2012

### d2j2003

How would I prove the derivative part of this? I know that (g(f(z)))'=g'(f(z))*f'(z) and obviously f' is not zero but i'm not sure how to show with certainty that g'(f(z)) is not zero...

5. Apr 8, 2012

### Dick

You are given that "the range of f a subset of the domain of g".

6. Apr 8, 2012

### HallsofIvy

Let P, Q, and R be three points such that angle PQR has measure $\theta$. Then, since f is conformal, P'=f(P), Q'=f(Q), and R'= f(R) are three points such that angle P'Q'R' has measure $\theta$. And, since g is conformal, P''= g(P'), Q''= g(Q'), and R''= g(R') are three points such that ....

7. Apr 8, 2012

### d2j2003

Such that the angle P''Q''R'' is θ meaning that the composition is angle preserving. right?

8. Apr 8, 2012

### d2j2003

can we say that since the range f(z) is in the domain of g then g(f(z)) is basically g(z)? and then it is non zero from there?

9. Apr 8, 2012

### Dick

f being conformal doesn't imply that. The 'angle preserving' in conformal refers to the angles between intersecting curves. Not to angles between points. That only works if f maps lines to lines.

10. Apr 8, 2012

### Dick

g' is nonzero on its domain. f(z) is in that domain. I wouldn't say 'g(f(z)) is basically g(z)', that's not very true.

11. Apr 8, 2012

### d2j2003

well when we say g(z), that z includes f(z) since the range of f is a subset of the domain of g... ie. g(z) takes values from its domain, which includes f(z).. right?

12. Apr 8, 2012

### Dick

Right. Just say f(z) is in the domain of g. Don't say g(f(z)) is the same as g(z).

13. Apr 8, 2012

### d2j2003

ok, got it. Thanks for your help