So the problem states that if $f(z)$ is entire, and satisfies the relation $f(z+i) = f(z)$ and $f(z+1) = f(z)$, show that $f(z)$ is constant. So I was thinking that since any point in $\mathbb{C}$ can be written as $\alpha * 1 + \beta * i $ we can say that $f(z + z_0) = f(z) $ in which case it is constant, but I'm having trouble breaking down the steps, and using the fact that f is entire, which makes me feel like I'm missing something. What should I review to figure this out?
How to Prove that if f(z) is entire, and f(z+i) = f(z), f(z+1) = f(z), then f(z) is constant?
10
$\begingroup$
complex-analysis
-
5Might try showing $f(\mathbb{C})$ is the same as $f(I\times I)$, where $I$ is the unit interval. Then of course since $I\times I$ is compact, the image is bounded, and Liouville saves the day. – 2011-05-04
-
4max modulus principle – 2011-05-04
-
0Thanks @yoyo, that was really helpful to review as well. It applies to another question that I was also stuck on. – 2011-05-04