(a) i. Find the modulus and amplitude of (1 + 2i)²(1 – i).

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.5: Applications
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(a) i. Find the modulus and amplitude of (1 + 2i)²(1 – i).
x2
y2
ii. If sin[log(A + iB)] = x + iy, show that
sin?u
1, where
cos?u
A² + B² = e2u.
(b) Prove that u = x² – y² + 2xy – 2x + 3y is harmonic. Find a function v
such that f (z) = u + iv is analytic. Also express f (z) in terms of z.
Transcribed Image Text:(a) i. Find the modulus and amplitude of (1 + 2i)²(1 – i). x2 y2 ii. If sin[log(A + iB)] = x + iy, show that sin?u 1, where cos?u A² + B² = e2u. (b) Prove that u = x² – y² + 2xy – 2x + 3y is harmonic. Find a function v such that f (z) = u + iv is analytic. Also express f (z) in terms of z.
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