DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passaudited 2026-08-11

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Improper integrals with several singular ends

Statement

If $f$ has a possible singularity at $c \in (a, b)$ , define $\int_{a}^{b} f : = \int_{a}^{c} f + \int_{c}^{b} f$ only when the two one-sided improper integrals converge separately. Cancellation between divergent sides is not allowed.

Likewise, for any finite split point $s$ , define $\int_{- \infty}^{\infty} f : = \int_{- \infty}^{s} f + \int_{s}^{\infty} f$ only when both tails converge. More generally, an interval with several singular ends is split into finitely many one-ended pieces, each of which must converge separately. Independence of the permitted split point is a theorem, not part of this definition.

Depends on

Dependency tree · 2 levels

Sources

William F. Trench, Introduction to Real Analysis, Section 3.4