to prove f is constant

• May 7th 2011, 12:19 PM
AcCeylan
to prove f is constant
Let f:C→C be differentiable, with f(z)/=0 (not equal) for all z in C. Suppose limz→z0 f(z) exist and is nonzero. Prove that f is constant.
• May 7th 2011, 12:25 PM
TheEmptySet
Quote:

Originally Posted by AcCeylan
Let f:C→C be differentiable, with f(z)/=0 for all z\inC. Suppose limz\to z0 f(z) exist and is nonzero. Prove that f is constant.

The statement is false consider the function

$f(z)=e^z \quad \lim_{z \to z_0}f(z)=e^{z_0}$ and this function is holomorphic, but is not constant.
• May 7th 2011, 12:37 PM
AcCeylan
Yes you are true!!?? but I found this question in a complex analysis book, I tried to solve but I hadnt thought that it could be false..!!
• May 8th 2011, 09:22 AM
Tinyboss
Can you get that f is an entire function whose image misses a neighborhood of zero? In that case, 1/f is a bounded entire function, and you can apply Liouville's.
• May 14th 2011, 04:59 AM
AcCeylan
Take limit e^z as z→-infinity it equal to zero, so it not satisfy the statement. I proved this statement with using Liouville Theorem