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Can anybody help me solving following problem?

Is there a sequence of polynomals P_n on complex plane such that

limit of P_n(z) equals 1 on upper half plane -1 on lower half plane and 0 on real axis?

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1 Answer

up vote 3 down vote accepted

Hint: Use Runge's theorem. Find sequences of compact sets $A_n$, $B_n$, $C_n$ so that

1) $A_n \subseteq A_{n+1}$, $B_n \subseteq B_{n+1}$, $C_n \subseteq C_{n+1}$

2) $\bigcup_n A_n$, $\bigcup_n B_n$, $\bigcup_n C_n$ are the upper half plane, lower half plane and real axis.

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I know that on each compact subset we can approximate uniformly by polynomials... But how can we sure that on overlapping regions these polynomials actually coincide? –  Detectives Nov 8 '12 at 14:18
    
For each $n$, you get one polynomial (not several that need to coincide) that approximates a certain function on the compact set $A_n \cup B_n \cup C_n$. –  Robert Israel Nov 8 '12 at 22:07
    
Thanks I got it. Especially that works with 1/n –  Detectives Nov 9 '12 at 3:30
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