Consider the following problems related to computation with complex

numbers.

(a) Find and plot all roots of

(i) z³ = − 1,           (ii) z² = 1               (iii) z² + 3z + 1 = 0

(b) Suppose you want to find the natural log of a complex number z = ∣z∣e^jθ, which is

log(z) = log(|z|e^jθ) = log(|z|) + log(e^jθ) = log(|z|) + jθ

If z is negative it can be written as z = ∣z∣e^jπ and we can find log(z) by using the above derivation. The natural log of any complex number can be obtained this way also. Justify each one of the steps in the above equation and find

(i) log( − 2),         (ii) log(1 + j1),    (iii) log(2e^jπ/4)

(c) Let z = −1 find (i) log(z), (ii) e^log(z).

(d) Let z = 2e^jπ/4 = 2(cos(π/4) + jsin(π/4))

(i) find z²/4,

(ii) is it true that(cos(π/4) + j sin(π/4))² = cos(π/2) + j sin(π/2) = j?