Summation Formula for Alternating Series over Half-Integers

Theorem

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

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

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

Suppose that:

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

as $N \to \infty$.

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

Let $Y$ be the set of poles of $\map f {\dfrac {2 z + 1} 2}$.

Then:

$\ds \sum_{n \mathop \in \Z \mathop \setminus Y} \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \mathop \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$

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

$\ds \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \mathop \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$

Proof

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Sources

• 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.8$: Summation of Series