Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-30
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FALSE: a meromorphic function always equals the naive sum of its principal parts

Statement

False claim: A meromorphic function is always the pointwise sum of its principal parts, with no convergence-forcing corrections.

Facts & Assumptions

Given: The principal parts pn(z)=−n/(z−n) at the positive integers.

[L1]

Mittag-Leffler on the plane needs correction terms to force convergence (Mittag-Leffler on the complex plane).

Refutation

technique · direct
1.1

Fix any noninteger positive real x. Then pn(x)=−n/(x−n)=1+x/(n−x)→1 as n→∞.

givenalgebra
2.1

Therefore the naive series ∑n≥1pn(x) has terms that do not tend to 0, so it diverges. This is exactly why the correction terms from [L1] are not optional in Mittag-Leffler's theorem.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources