Past exam of the mathematical methods I course of the University of Oxford / 2017 / 1

Since $(1+i{)}^2=2i$, the two square roots are $1+i$ and $-1-i$. The principal square root is $1+i$, whose real part isIf the notation is meant to include both square-root values, their real parts are $+1$ and $-1$. Stating the principal-branch convention removes that ambiguity.

For $\delta \notin 2\pi \mathbb{Z}$, the finite geometric series givesThe exponential prefactor has modulus one. ThusAt an excluded phase every summand is one, which supplies the removable limiting value directly. Here $N$ is a positive integer.

From the exponential definition, $\cos (iu)=\cosh u$. ThereforeA cosine of a nonreal argument need not lie between $-1$ and $1$; that bound applies to real arguments.