Problem 1744 (difficulty: 5/10)

Let \(\displaystyle a\in\CC\), \(\displaystyle |a|<1\), \(\displaystyle n\in\NN\). Show that \(\displaystyle (z-1)^ne^z=a\) has exactly \(\displaystyle n\) solutions in the half-plane \(\displaystyle \re z>0\).

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