Summation Formula (Complex Analysis)

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

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

Let $f$ be a meromorphic function on $\C$ with finitely many poles.

Suppose that:


 * $\ds \int_{C_N} \paren {\pi \cot \pi z} \map f z \rd z \to 0$

as $N \to \infty$.

Let $X$ be the set of poles of $f$.

Then:
 * $\ds \sum_{n \mathop \in \Z \mathop \setminus X} \map f n = - \sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0}$

If $X \cap \Z = \O$, this becomes:


 * $\ds \sum_{n \mathop = -\infty}^\infty \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0}$

Proof
By Summation Formula: Lemma, there exists a constant $A$ such that:


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

for all $z$ on $C_N$.

Since $f$ has only finitely many poles, we can take $N$ large enough so that no poles of $f$ lie on $C_N$.

Let $X_N$ be the set of poles of $f$ contained in the region bounded by $C_N$.

From Poles of Cotangent Function, $\map \cot {\pi z}$ has poles at $z \in \Z$.

Let $A_N = \set {n \in \Z : -N \le n \le N}$

We then have:

We then have, for each integer $n$:

Note that by hypothesis:


 * $\ds \int_{C_N} \paren {\pi \cot \pi z} \map f z \rd z \to 0$

So, taking $N \to \infty$:


 * $\ds 0 = 2 \pi i \paren {\sum_{n \mathop \in \Z \mathop \setminus X} \map f n + \sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0} }$

which gives:


 * $\ds \sum_{n \mathop = -\infty}^\infty \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0}$