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    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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