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1. Prove each of the following results: (a) |z − w|=|w-z, Vw, z = C (b) ||z − w|| ≤|zw|, Vw, z C - (c) |z| ≤ Re(z) ≤ 2, V₂ € C (d) e² = e, Vze C. 2. Simplify completely:

1. Prove each of the following results: (a) |z − w|=|w-z, Vw, z = C (b) ||z − w|| ≤|zw|, Vw, z C - (c) |z| ≤ Re(z) ≤ 2, V₂ € C (d) e² = e, Vze C. 2. Simplify completely:

Advanced Engineering Mathematics
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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1. Prove each of the following results: (a) |zw|=|w-z, w, z C (b) ||z|-|w|| ≤|zw| vw, z (c) 2 ≤ Re(z) ≤ 2, VEC (d) e² = e, Vze C. 2. Simplify completely: (a) √√1+i√3 (b) (2+3i)¹+i Part 2. : 1. Describe and sketch each of the following sets of complex numbers: (a) 1 < |2z - 8| < 6 (b) |z-2-i = 3 (c) Re(iz + 2) >0 (d) |z1|²+ |z+ i|² < 2. 2. : (a) For n ≥ 1, show that 1+z+z² 1 - ₂n+1 1-2 (b) Hence, show that 1 + cos 0 + cos 20 + ... + cos n0 = 1/2+ +2¹= 1 21 sin(n+¹)0 2 sin /
Transcribed Image Text:1. Prove each of the following results: (a) |zw|=|w-z, w, z C (b) ||z|-|w|| ≤|zw| vw, z (c) 2 ≤ Re(z) ≤ 2, VEC (d) e² = e, Vze C. 2. Simplify completely: (a) √√1+i√3 (b) (2+3i)¹+i Part 2. : 1. Describe and sketch each of the following sets of complex numbers: (a) 1 < |2z - 8| < 6 (b) |z-2-i = 3 (c) Re(iz + 2) >0 (d) |z1|²+ |z+ i|² < 2. 2. : (a) For n ≥ 1, show that 1+z+z² 1 - ₂n+1 1-2 (b) Hence, show that 1 + cos 0 + cos 20 + ... + cos n0 = 1/2+ +2¹= 1 21 sin(n+¹)0 2 sin /
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