Let f be analytic on a domain D and suppose that f'(x) = 0 for all z e D. Show that f is constant on D.
Can someone show how to show this please. I have an exam tomorrow and I need to know how to do this...
Thanks in advance for any help
Let f be analytic on a domain D and suppose that f'(x) = 0 for all z e D. Show that f is constant on D.
Can someone show how to show this please. I have an exam tomorrow and I need to know how to do this...
Thanks in advance for any help
It's a little trickier than the real-variable problem, where you could just use the mean-value theorem.
Here the trick is to fix z_0 in D (which you have to assume is connected) and let w_0 = f(z_0). Let A = {z in G | f(z) = w_0}. You can show A = D by showing that A is both open and closed in D. The details are messy... I recall there is a pretty good proof in Conway though.
You may find this thread informative.