- #1
Bacle
- 662
- 1
Esteemed Analysts:
I am trying to rigorize the result that if f is 1-1 in a region R, then f'(z) is not zero in R.
This is what I have: Assume, by contradiction, that f'(zo)=0 for zo in R. Then
f can be expressed locally as :
f(z)=z^k.g(z)
for g(z) analytic and non-zero for some open ball B(zo,r)-{zo}
From this, we have to somehow use the fact that z^k is k-to-1, contradicting the
assumption that f is 1-1.
I don't know if we can use the fact that an open ball is simply-connected to
define a branch of log, from which we can define a branch of z^k, and then
conclude with the contradiction that f(z) is not 1-1.
Any suggestions for rigorizing?
Thanks.
I am trying to rigorize the result that if f is 1-1 in a region R, then f'(z) is not zero in R.
This is what I have: Assume, by contradiction, that f'(zo)=0 for zo in R. Then
f can be expressed locally as :
f(z)=z^k.g(z)
for g(z) analytic and non-zero for some open ball B(zo,r)-{zo}
From this, we have to somehow use the fact that z^k is k-to-1, contradicting the
assumption that f is 1-1.
I don't know if we can use the fact that an open ball is simply-connected to
define a branch of log, from which we can define a branch of z^k, and then
conclude with the contradiction that f(z) is not 1-1.
Any suggestions for rigorizing?
Thanks.