Homework 4
Hint for IV.6.3 --Steve Bell
We assume $ (f)''=f $ on $ \mathbb C $.
Notice that
$ (e^z f)''=e^zf +2e^zf'+e^zf''=2(e^zf + e^zf')=2(e^zf)'. $
Let $ g=e^zf. $ Then $ g'=2g $ and now you can use the theorem from class that concerns solutions of this first order complex ODE. By the way, you will also need to use the fact that if two analytic functions on the complex plane have the same derivative, then they must differ by a constant.
Could you just substitute the f in the form they provided, into the ODE to prove that you can assume the solution is of that form? --Adrian Delancy
