Summation Formula (Complex Analysis)/Lemma

Theorem
Let $N \in \N$ be an arbitrary natural number.

Let $C_N$ be the square embedded in the complex plane with vertices $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$.

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


 * $\cmod {\map \cot {\pi z} } < A$

for all $z \in C_N$.

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 $A = \map \max {A_1, A_2}$ gives the desired bound.