Complex Analysis: Theory and Applications · Problem 4.1.8

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · polar-form-sequence-limits

Problem

Let zn=rneiθn≠0z_n = r_n e^{i\theta_n} \neq 0, where θn=Arg⁡(zn)\theta_n = \operatorname{Arg}(z_n). (a) Suppose lim⁡n→∞rn=r0\displaystyle\lim_{n\to\infty} r_n = r_0 and lim⁡n→∞θn=θ0\displaystyle\lim_{n\to\infty} \theta_n = \theta_0. Show that lim⁡n→∞rneiθn=r0eiθ0\displaystyle\lim_{n\to\infty} r_n e^{i\theta_n} = r_0 e^{i\theta_0}. (b) Find an example of a sequence {zn}={rneiθn}\{z_n\} = \{r_n e^{i\theta_n}\} where lim⁡n→∞zn=z0=r0eiθ0\displaystyle\lim_{n\to\infty} z_n = z_0 = r_0 e^{i\theta_0} and lim⁡n→∞rn=r0\displaystyle\lim_{n\to\infty} r_n = r_0, but lim⁡n→∞θn\displaystyle\lim_{n\to\infty} \theta_n does not exist. (c) If {zn}={rneiθn}\{z_n\} = \{r_n e^{i\theta_n}\}, is it possible to have lim⁡n→∞zn=z0=r0eiθ0\displaystyle\lim_{n\to\infty} z_n = z_0 = r_0 e^{i\theta_0}, but lim⁡n→∞rn\displaystyle\lim_{n\to\infty} r_n does not exist?

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