I am working on old qualifying exams, and I have stumbled on a problem that looks somewhat like this:
Prove that $f(z) = (z-1)^{50}e^z + .75(z+1)^{50}$ has exactly 50 simple zeroes with $\Re z > 0 $
(The actual problem is more general)
Now, proving the existence of 50 zeroes in the right plane is just a matter of Rouche's Theorem, that is clear to me. What I am unsure about is how to prove the simple part -- I have never seen a problem ask to prove this. Rouche's Theorem, as I understand, provides no information about the multiplicity of zeroes. The only theorems I know of that deal with multiplicity of zeroes is the Argument Principle or the Residue Theorem. Only the Argument Principle looks promising, but just toying with it, it seems impractical to implement.
Is there a standard method of proof or theorem that I am missing to prove something like this for a general function? Any hints on this particular problem would be appreciated!