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Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation

$\frac{\partial}{\partial z} f = \frac{\partial}{\partial \bar{z}} f = \frac{i}{2r} f$

where $r$ is the radius of $\mathbb{CP}^1$, have nontrivial solutions? If yes, then is there an easy example?

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  • $\begingroup$ Well, I guess I should say nontrivial solutions. $\endgroup$
    – Moduli
    Dec 24, 2012 at 23:43

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It sounds like the only non-trivial solution is $ae^{i(z+\bar z)/2r}=ae^{-y/r}$. Taking difference of two equations we see that all solution depend only on $y$ and if you restrict to $y$ axis the equation is easy to solve.

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  • $\begingroup$ Indeed, this seems to be the only nontrivial solution. $\endgroup$
    – Moduli
    Dec 25, 2012 at 0:35
  • $\begingroup$ Wait, $z+\bar{z}=2x$, so the solutions is $ae^{i(z+\bar{z})/2r}=ae^{ix/r}$, which is ill defined at $x\rightarrow \infty$. $\endgroup$
    – Moduli
    Dec 28, 2012 at 20:57
  • $\begingroup$ Sure, this solution is ill defined at $x\to \infty$. In over words there are no solutions defined on whole $\mathbb CP^1$, a solution can only be defined on $\mathbb C$. $\endgroup$
    – Dmitri Panov
    Dec 28, 2012 at 21:38

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