Proving Entire Functions are Constant: f, e^f, Re f

[mathusers]

let f be an entire function..

1) prove if e^f is bounded then f is constant
2) prove that if Re f is bounded then f is constant

i'm guessing you would have to use suitable exponentials but i don't have a good enough idea of what to do here. any help would be greatly appreciated :Dxx

 

[Office_Shredder]

You can start with if an entire function is bounded everywhere, it's constant.

http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)

I think the proof moves the absolute value bars into the integral a bit prematurely (as generally a path integral of a real function can give a complex number as an answer, but I can't be bothered to check if in this case it actually comes out real all the time), but the basic idea is there