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

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.

The polar form $1+i=\sqrt{2}{e}^{i\pi /4}$ gives $(1+i{)}^6=-8i$. Multiplication therefore scales the modulus by $8$ and rotates the complex argument through $-\pi /2$:Equivalently the result is $8b-8ia$, whose complex argument is $atan2(-a,b)$. If $a=b=0$, its modulus is zero and its complex argument is undefined.