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Math Help - Solutions of complex equations

  1. #1
    Junior Member
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    Solutions of complex equations

    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.
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  2. #2
    Super Member Rebesques's Avatar
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    Number 2 is actually Rouche's Theorem. Use it to prove the other two as follows. One side is f, the other is g.
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