Number 2 is actually Rouche's Theorem. Use it to prove the other two as follows. One side is f, the other is g.
I missed the class where we discussed this, could somebody help me with these three problems?
1. Prove that e^z = 5z^3 - 1 has 3 solutions in |z|< 1.
2. Let f and g be analytic on a domain containing a simple closed curve C and its inside. Show that if |f(z)|>|g(z)| for all z in C, then the two equations f(z) = g(z) and f(z) = 0 have an equal number of solutions inside C.
3. Determine the number of solutions of z^3 -10 = z^2 + z + 6 in |z|< 1.
Any help would be greatly appreciated.
Number 2 is actually Rouche's Theorem. Use it to prove the other two as follows. One side is f, the other is g.