# Complex Variables

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If a > e prove that the equation a*z^n=e^z has n solutions (counting multiplicities) inside of the circle |z|=1.

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#### Solution Preview

Proof. Let f(z)=a*z^n and g(z)=-e^z. On the circle C: |z|=1, we have

|f(z)|=|a*z^n|=a

|g(z)|=|-e^z|<=e

Since ...

#### Solution Summary

This proves that a given equation has n solutions inside of a given circle.

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