Complex Analysis: Theory and Applications · Problem 1.3.4

Complex Numbers — Howell & Matthews

Mathematics — Complex Analysis · modulus-identities-proofs

Problem

Prove the following Identities for z=x+iyz = x + iy: (a) Identity (1.21): ∣x∣=∣Re(z)∣≤∣z∣|x| = |\text{Re}(z)| \leq |z| and ∣y∣=∣Im(z)∣≤∣z∣|y| = |\text{Im}(z)| \leq |z|. (b) Identity (1.22), the triangle inequality: ∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2|. (c) Identity (1.26): ∣z1z2∣=∣z1∣∣z2∣\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}, provided z2≠0z_2 \neq 0.

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