- (10 points) How many solutions does the equation
\[ e ^ { z } = 3 z ^ { 7 } \]
have in the unit disk \(D = \{ x \in \mathbb { C } : | z | < 1 \} ?\) Justify your answer.
Solution
1. On \(\abs z=1\), \(\abs{e^z}<\abs{-3z^7}\).
\(\abs{e^z}=e^{\Re z}\le e<3=\abs{-3z^7}\).
2. Q.E.D.
By step 1 and Rouché’s theorem, \(e^z-3z^7\) has as many zeros in \(\abs z<1\) as \(-3z^7\), counted with multiplicity, namely \(\boxed{7}\). Counted without multiplicity the answer is the same: at a zero, \(e^z=3z^7\) and the derivative \(e^z-21z^6=3z^7-21z^6=3z^6(z-7)\) is nonzero for \(0<\abs z<1\), and \(z=0\) is not a zero.