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I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints?

For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ with poles at $\infty$ such that $f^n+g^n =1$.

Thomas Andrews
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mathgenesis22813
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  • there are entire functions $a,b,c,d$ such that $f = a/c$ and $g = b/d$, and $f,g$ have their poles at the same points and of same order, so without any loss of generality we can suppose that $c = d$ ? this would reduce the problem to $a^n + b^n = c^n$ where $a,b,c$ are entire functions. – reuns Feb 01 '16 at 17:21

1 Answers1

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From I. N. Baker, On a class of meromorphic functions. Link to the paper in JStor.

For $n=2$:

All solutions are of the form $$\begin{align}f(z)&=\frac{2h(z)}{h^2(z)+1}\\g(z)&=\pm\frac{h^2(z)-1}{h^2(z)+1}\end{align}$$ for $h(z)$ any meromorphic function.

For $n=3$:

All solutions are of the form $$\begin{align}f(z)&=\frac{\frac{1}{2}+\frac{\wp'(h(z))}{\sqrt{12}}}{\wp(h(z))}\\g(z)&=\omega\frac{\frac{1}{2}-\frac{\wp'(h(z))}{\sqrt{12}}}{\wp(h(z))}\end{align}$$

where $\wp(z)$ is Weierstrass elliptic function, for $g_2=0$ and $g_3=1$, and $\omega$ is a cubic root of $1$.

For $n>3$:

There are no meromorphic solutions.