(a) Let z, w be two complex numbers such that zw 1. Prove thatWP1 - wzand also thatZW1 - wz2< 1 if |z| < 1 and |w| < 1,= 1if |z| = 1 or |w| = 1.[Hint: Why can one assume that z is real? It then suffices to prove that(r − w) (r − w) ≤ (1 − rw)(1 – rw)---with equality for appropriate r and w

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Answered: (a) Let z, w be two complex numbers…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(a) Let z, w be two complex numbers such that zw 1. Prove that w z Wz and also that w z 1 - wz < 1 1 if |2 < 1 and |w| < 1, if |z| = 1 or |w| = 1. [Hint: Why can one assume that z is real? It then suffices to prove that (rw) (rw) ≤ (1-rw)(1-rw) with equality for appropriate r and wl.]