Math 207 HW4: Problems not from Stein and Shakarchi
1. Use Rouche’s theorem to establish the fundamental theorem of algebra.
2. Find all solutions of the equation ez= 1 + 2zthat satisfy |z|<1.
3.Fix a complex number λ, with |λ|<1. For n≥1 .Show that
(z−1)nez−λhas n zeros satisfying |z−1|<1
and no other zeros on the right hand half-plane. Determine the multiplicity
of the zeros.
3. If z0is a isolated singularity that is not removable of f(z) . Show that
z0is an essential singularity of ef(z).
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