Complex Analysis Answers

Simplify the following expressions: (a) (cos π/4 + i sin π/4)(cos 3π/4 + i sin 3π/4), (b)

√ 3, (b) w = −2 1 √

(cos π/4 + i sin π/4)2 (cos π/6 + i sin π/6)

. Find the modulus and the principal argument of each of the given complex numbers. (a) 3 − 4i, (b) −2 + i, (c)

1 1 + i √ , (d) 3 7 − i −4 − 3i (e) 5(cos π/3 + i sin π/3), (f) cos 2π/3 − sin 2π/3

Given that z = 1 + i √ 2, express in the form a + ib each of the complex numbers

p = z + 1/z, q = z − 1/z. In an Argand diagram, P and Q are the points which represent p and q respectively, O is the orgin, M is the midpoint of PQ and G is the point on OM such that OG =

2 3

OM. Prove that angle PGQ is a right angle.

given that w denotes either one of the non real roots of the equation z³=1. show that a. 1+w+w²=0 b. the other non real rootvis w². show that the non real root of the equation ((1-u)÷u)² can be expeessed in the form Aw and Bw², where A and B are real numbers and A and B

Express the roots of the equation z3 −α3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib. a) z^2-27=0

Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 − 2i| and |z − 4i| = |z + 2i| simultaneously.

2. (a) Find the real root of the equation z3 + z + 10 = 0 given that one root is 1 − 2i.

Show that z = i is a root of the equation z4 + z3 + z − 1 = 0. Find the three

other roots.

(a) Find the real root of the equation z3 + z + 10 = 0 given that one root is 1 − 2i.

(b) Given that 3 + i is a root of the equation z3 − 3z2 − 8z + 30 = 0, find the

remaining roots.

Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 − 2i| and |z − 4i| = |z + 2i| simultaneously

. Express the roots of the equation z3 − α3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib.