CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The partial-fraction expansion of pi-squared cosecant-squared

Statement

For every $z \in C ∖ Z$ ,

$π^{2} \csc^{2} (π z) = \sum_{n \in Z} \frac{1}{(z - n)^{2}} = \frac{1}{z^{2}} + \sum_{n \geq 1} (\frac{1}{(z - n)^{2}} + \frac{1}{(z + n)^{2}}) .$

with locally uniform convergence on $C ∖ Z$ .

Facts & Assumptions

Given: The cotangent expansion on $C ∖ Z$ .

[L1]

On $C ∖ Z$ , $π \cot (π z) = 1 / z + \sum_{n \geq 1} 2 z / (z^{2} - n^{2})$ (The Mittag-Leffler expansion of pi cotangent).

[L2]

The derivative of $\cot$ is $- \csc^{2}$ (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives).

[L3]

Locally uniform convergence of holomorphic partial sums forces local-uniform convergence of their derivatives to the derivative of the limit (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

Proof

technique · direct

1.1

Let $S_{N} (z) : = \frac{1}{z} + \sum_{n = 1}^{N} (\frac{1}{z - n} + \frac{1}{z + n}) .$ By [L1], the holomorphic functions $S_{N}$ converge locally uniformly on $C ∖ Z$ to $π \cot (π z)$ . On any compact set $K \subseteq C ∖ Z$ , the derivatives satisfy $S_{N}^{'} (z) = - \frac{1}{z^{2}} - \sum_{n = 1}^{N} (\frac{1}{(z - n)^{2}} + \frac{1}{(z + n)^{2}}),$ and this derivative series converges uniformly on $K$ because its terms are $O (n^{- 2})$ there. Therefore [L3] identifies the derivative of the limit with the displayed series on $K$ .

L1L3given

2.1

Differentiating [L1] gives $- π^{2} \csc^{2} (π z) = - \frac{1}{z^{2}} - \sum_{n \geq 1} (\frac{1}{(z - n)^{2}} + \frac{1}{(z + n)^{2}})$ by [L2]. Multiplying by $- 1$ yields the claimed formula.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Weber, Complex Analysis, Example 3.3.1 (standard reference, not scraped)
J. Lebl, Guide to Cultivating Complex Analysis, §9.4 (standard reference, not scraped)