Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 1 / i / Solution

The complex exponential factors as $e^{(3+i)(a+ib)}=e^{3a-b}{e}^{i(a+3b)}$. Its positive real prefactor is its modulus, while its phase gives its complex argument:For the principal complex argument, reduce $a+3b$ to the chosen principal interval.