Complex Analysis: Theory and Applications · Problem 4.2.1

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · newtons-method-complex

Problem

Consider the function f(z)=z2+1f(z) = z^2 + 1, where N(z)=z−f(z)f′(z)=z2−12z=12(z−1z).N(z) = z - \frac{f(z)}{f'(z)} = \frac{z^2-1}{2z} = \frac{1}{2}\left(z - \frac{1}{z}\right). (a) Show that, if Im⁡(z0)>0\operatorname{Im}(z_0) > 0, the sequence {zk}\{z_k\} formed by successive iterations of z0z_0 via N(z)N(z) lies entirely within the upper half-plane. (b) Show that a similar result holds if Im⁡(z0)<0\operatorname{Im}(z_0) < 0. (c) Use induction to show that, if all the terms of the sequence {zk}\{z_k\} are defined, then the sequence {zk}\{z_k\} is real, provided z0z_0 is real. (d) Discuss whether {zk}\{z_k\} converges to ii if Im⁡(z0)>0\operatorname{Im}(z_0) > 0 and to −i-i if Im⁡(z0)<0\operatorname{Im}(z_0) < 0.

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