I have been trying the some exercises given in the book of "Robert E green and Krantz".But I am unable to do it,the exercise is following

Construct a sequence of entire function $f_j$ such that $f_j$ converges uniformly to $1$ on compact subsets of open right half plane but $f_j$ does not converge to any point of open left half plane

I am thinking to do it as an application of Runge's Approximation theorem that state as "If f is holomorphic on a neighborhood of compact set $K$ such that $\mathbb{C}\backslash K$ is connected then $f$ can be approximated by polynomials uniformly on $K$ .

I know Polynomials are entire functions but converse is not true for example exponential functions that are entire but not polynomials. So My question is how this theorem helps me to do this question.For example if am able to find the sequence of polynomials then how can be sure this is ok? Please help me in understanding this question.

  • $\begingroup$ You have to assume that $\mathbb C \setminus K$ is connected (although $K$ itself may be disconnected) . This is what David alludes to when he writes (in his optimally simple solution below) about an incorrect statement of Runge's theorem . $\endgroup$ – Georges Elencwajg Jan 10 '18 at 21:19

Your statement of Runge's theorem is false! You're leaving out some important hypotheses.

Anyway, if $R_n$ is an increasing sequence of rectangles with union equal to the right half plane and $R_n'$ is an increasing sequence of rectangles with union equal to the left half plane you can use a correct version of Runge's theorem to find polynomials $P_n$ so $|P_n-1|<1/n$ on $R_n$ and $|P_n-(-1)^n|<1/n$ on $R_n'$.

  • $\begingroup$ Jawoll ! ${}{}$ $\endgroup$ – Georges Elencwajg Jan 10 '18 at 21:23
  • $\begingroup$ @David C.Ullrich Sir thanks for giving me nice hint. I got that in this way we can find sequence of polynomials that converge to $1$ uniformly on right half plane.But I am unable to get that how this implies that it will not converge to any point of left half plane. Please can you elaborate this more. $\endgroup$ – John Jan 11 '18 at 10:05

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