Summation Formula for Alternating Series over Half-Integers
Jump to navigation
Jump to search
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
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 2009: Murray R. Spiegel, Seymour Lipschutz, John Schiller and Dennis Spellman: Complex Variables (2nd ed.): $7.8$: Summation of Series