 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/1/ii/solution

1 + i = 2 e^{iπ /4}

gives

(1 + i)^{6} = - 8 i

. Multiplication therefore scales the modulus by

8

and rotates the complex argument through

- π /2

∣ (1 + i)^{6} z ∣ = 8 a^{2} + b^{2}, ar g ((1 + i)^{6} z) = ar g (a + ib) - \frac{π}{2} (mod 2 π) .

Equivalently the result is

8 b - 8 ia

atan2 (- a, b)

. If

a = b = 0

, its modulus is zero and its complex argument is undefined.

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