Complex Analysis Answers

4. If the roots of the cubic az3 + bz2 + cz +d = 0 form an arithmetic progression α−β, α, α + β, prove that (2b2 − 9ac)b + 27a2d = 0.

Express sin 5θ and cos 5θ/ cos θ in terms of sin θ

Express −1 + i in polar form. Hence show that (−1 + i)16 is real and that 1/(−1 + i)6

is purely imaginary, giving the value for each.

Use De Moivre’s theorem to simplify the following

(a) (cos π/5+i sin π/5)10, (b) (cos π/9+i sin π/9)−3, (c) {cos(−π/6) + i sin(−π/6)}−4

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.

9.The principal value of the argument of (1+ ˆš3)(1−i) is____

\\(-5\\fra{\\pi}{12}\\)

\\(\\fra{5\\pi}{12}\\)

\\(\\fra{-\\pi}{12}\\)

\\(\\fra{\\pi}{12}\\)

Show that w=z+e^z is analytic and hence find dw/dz

9.3 Let w be a negative real number, z a 6

th root of w.

(a) Show that z (k) = ρ^

1

6

-


cos (pi+2kpi/6 )+ isin ( pi+2kpi/6), , k = 0, 1, 2, 3, 4, 5 is a formula for the

6th roots of w.

Given z = cos θ + isin θ and u + iv = (1 + z)(1 + z

2

). Prove that v = u tan( 3θ

2

) and

u

2 + v

2 = 16 cos2

(

θ

2

) cos2

(θ)

Solve (-1)^2i