Prove that $e^{f_1} + e^{f_2} \equiv 1$ is impossible

Asked Apr 29 '23 at 05:31

Active Apr 29 '23 at 05:31

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Problem: Let $f_1, f_2$ be two entire functions not constant. Prove that $e^{f_1} + e^{f_2} \equiv 1$ is impossible.

I tried to apply the Picard's theorem to investigate the left side of $e^{f_1} + e^{f_2} \equiv 1$, but fail to gain any progress. If we try to represent $f_1$ or $f_2$ by the other one, the multiplicity is inevitable.

asked Apr 29 '23 at 05:31

Jech

6

Does this answer your question? Complex Analysis, Entire functions (found using Approach$0$) – Anne Bauval Apr 29 '23 at 05:36

Prove that $e^{f_1} + e^{f_2} \equiv 1$ is impossible

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