M417

Fall 1996

hw11.tex due Nov 6, 1996

Let f(z) have an isolated singularity at z = 0.

1.: Suppose that |f(z)| is bounded away from 0 for z near 0; i\. e\. , there are R > 0, δ > 0, such that for 0 < |z| < R, f(z) is analytic and |f(z)| > δ. Then f does not have an essential singularity at z=0.

2.: If f has an essential singularity at z = 0, and w₀ is any complex number, then there is a sequence {z_n} such that lim_{n → ∞}z_n = 0 and lim_n→∞f(z_n)=w₀.

3.: If lim_{z → 0}|f(z)| = ∞, then f has a pole at z=0, but f(z) does not have an essential singularity at z = 0.

One can now show the following result:

Theorem Suppose that g(z) is an entire function such that lim_{z → ∞}g(z) = ∞. Then g(z) is a polynomial.
The function f(z) = g([1/z]) satisfies the hypotheses of the third exercise. For z near 0, f(z) = ∑_n=0^Nb_n z⁻ⁿ and g(z) = ∑_n=0^Nb_n zⁿ. Why are there no other nonzero terms?

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On 06 Feb 2017, 13:23.