# Bounded Function

1. Dec 4, 2008

### alvielwj

How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C.
π is pi.

2. Dec 4, 2008

### HallsofIvy

Staff Emeritus
How to show "if ...."

but where is your conclusion? What do you want to prove?

3. Dec 4, 2008

### alvielwj

need to prove f(z) is constant.
first show f is bounded,then by the Liouville's theorem, f is constant

4. Dec 4, 2008

### alvielwj

let me post the whole question
Suppose that f is an entire function such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C. Use Liouville's theorem to show that f is constant.
Hint: Consider the restriction of f to the square {z = x + iy : 0 <x < 2π ; 0 < y <2π }

5. Dec 5, 2008

### HallsofIvy

Staff Emeritus
Looks like a good hint! Although wasn't it $0\le x\le 2\pi$, $0\le y\le 2\pi$? The "=" part is important because that way the set is both closed and bounded and so any continuous function is bounded on it. Since f is "periodic" with periods $2\pi$ and $2\pi i$, the bounds on that square are the bounds for all z.

6. Dec 5, 2008