Summation Formula (Complex Analysis)/Lemma

Theorem
Let $C_N$ be the square with vertices $\left({N + \frac 1 2}\right) \left({\pm 1 \pm i}\right)$ for $N \in \N$.

Then there exists a constant $A$ independent of $N$ such that:


 * $\displaystyle \left\vert{\cot \left({\pi z}\right)}\right\vert < A$

for all $z$ on $C_N$.

Proof
Let $z = x + iy$ for real $x, y$.

Case 1: $y > \frac 1 2$
We have:

Case 2: $y < -\frac 1 2$
Similarly:

Case 3: $-\frac 1 2 \le y \le \frac 1 2$
First consider $z = N + \frac 1 2 + iy$.

Then:

Similarly in the case of $z = -N - \frac 1 2 + iy$, we have:

Picking $A = \max \left({A_1, A_2}\right)$ gives the desired bound.