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I have some problem on finding roots and showing that they exist under certain conditions:

Problem: Prove that $e^z+z^3$ has no root in $\{z:|z|<3/4\}$ and has three roots in $\{z:|z|<2\}.$

I am preparing for my finals for complex analysis, and this is one of the review problems, but I cannot seem to solve it and its frustrating as I am stuck on this chapter complex roots and residues. I would appreciate if someone could show me how to solve this.

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Hint: For the first part, consider the possible values of $|e^z|$ and $|z^3|$ in that disk. For the second part, use Rouche's Theorem.

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  • $\begingroup$ Sorry I am not sure, I still am quite stuck at showing $3$ roots exist in $|z|<2$. $\endgroup$ – Aurora Borealis Apr 18 '18 at 8:23
  • $\begingroup$ @AuroraBorealis Are you familiar with Rouche's Theorem? $\endgroup$ – BallBoy Apr 18 '18 at 10:40
  • $\begingroup$ In my text I cannot find Rouche's theorem, so I tried to look it up and studying it now, but I am not sure how to apply this theorem to this problem. $\endgroup$ – Aurora Borealis Apr 18 '18 at 12:44
  • $\begingroup$ @AuroraBorealis There are a bunch of examples on this site applying Rouche's Theorem to this type of problem; look at math.stackexchange.com/questions/773276/… and math.stackexchange.com/questions/1162675/… as examples. You want to express your function as a sum $f(z)+g(z)$, where you know how many zeroes $f$ has in $|z|<2$, and $|g(z)|<|f(z)|$ on the boundary $|z|=2$. $\endgroup$ – BallBoy Apr 18 '18 at 16:00
  • $\begingroup$ Hmmm Ok I will take a look at them and try to solve it. Thank you. $\endgroup$ – Aurora Borealis Apr 19 '18 at 2:18

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