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 xx
how can you say if f is entire then so is e^f... is this just an assumption or is there a proof for this...
also, if the result follows from louville's theorem, are we meant to show the taylor series for e^f about 0??? where would we go from there.. i would be greatful if you could show me what is to be done here???
I suppose this just amounts to the chain rule from calculus. If f is differentiable then so is e^f, with . The definition of an entire function is that it is a function that is everywhere differentiable, and it follows that if f is entire then so is e^f.
Liouville's theorem says that if an entire function is bounded then it is constant. If e^f is constant then so is f. You don't need to bring the Taylor series into it.