Complex Analysis: Theory and Applications · Problem 8.7.8

Conformal Mapping — Howell & Matthews

Mathematics — Complex Analysis · rouche-theorem-fundamental-theorem-algebra

Problem

Use Rouch\'e's theorem to prove the fundamental theorem of algebra. Hint: For the polynomial g(z)=a0+a1z+⋯+an−1zn−1+anzng(z) = a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} + a_n z^n, let f(z)=−anznf(z) = -a_n z^n. Show that, for points zz on the circle CR(0)C_R(0), ∣f(z)+g(z)f(z)∣<∣a0∣+∣a1∣+⋯+∣an−1∣∣an∣R,\left|\frac{f(z) + g(z)}{f(z)}\right| < \frac{|a_0| + |a_1| + \cdots + |a_{n-1}|}{|a_n| R}, and conclude that the right side of this inequality is less than 1 when RR is large.

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