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could any one give me a hint how to show Every rational function which is holomorphic on every point of Riemann Sphere( $\mathbb{C}_{\infty}$) must be constant?(with out applying Maximum Modulas Theorem). Thank you.

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Does using the fact that the Riemann sphere is compact count as using the maximum modulus principle? –  Zhen Lin Jul 24 '12 at 11:55
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1 Answer 1

If a function is analytic on the sphere at $\infty$, it is bounded in an neighborhood of $\infty$. Consequently, it is bounded globally, since the complement of a neighbhorhood at $\infty$ is compact. Now invoke Liouville's theorem; the functon must be constant.

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why it is bounded in an neighborhood of $\infty$? –  Bunuelian Trick Jul 24 '12 at 13:42
    
If it is analytic at $\infty$, it is continuous there and it therefore bounded there. Note that a polynomial has a pole at $\infty$ on the Riemann sphere. –  ncmathsadist Jul 24 '12 at 14:38
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