 Solution

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

e^{(3 + i) (a + ib)} = e^{3 a - b} e^{i (a + 3 b)}

. Its positive real prefactor is its modulus, while its phase gives its complex argument:

∣ e^{(3 + i) z} ∣ = e^{3 a - b}, ar g (e^{(3 + i) z}) = a + 3 b (mod 2 π) .

For the principal complex argument, reduce

a + 3 b

to the chosen principal interval.

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