Show Boundedness of Entire Function f: f(z) = f(z + 2π ) & f(z + 2π i)

[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.

 

[HallsofIvy]

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.

How to show "if ..."

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

 

[alvielwj]

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

 

[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 }

 

[HallsofIvy]

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.

 

[alvielwj]

Thank you for your answer..
I finally know how to use the hint..
At the beginning i really don't know how to start..