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Suppose we have entire functions $f$ and $g$ that satisfy $e^{f(z)} + e^{g(z)} = 1 $ for all complex values $z$. Show that $f$ and $g$ are constant.

Jonas Meyer
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wellfedgremlin
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  • What tools have you to work with? Are you a beginner, or advanced? – Daniel Fischer Jan 03 '14 at 22:12
  • I familiar with one semester of graduate complex analysis. I've tried looking at properties after differentiating this equation. And I've tried taking a look at the at what happens at infinity. – wellfedgremlin Jan 03 '14 at 22:16

1 Answers1

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I take it

one semester of graduate complex analysis

means you have heard of Picard's so-called "little" theorem.

The functions $e^{f(z)}$ and $e^{g(z)}$ are entire, and omit two values, hence they are constant. That immediately implies that $f$ and $g$ are constant.

Daniel Fischer
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    Hey that's slick, I even thought about using that but then missed the fact that each can't take value 1. Thanks for your help! – wellfedgremlin Jan 03 '14 at 22:24