Alphabeta Math

Improper Integrals

About this page
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.

✓ 25 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 25 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Improper integrals over unbounded intervals

Definition

Improper integrals at a finite singular endpoint

Definition

Improper integrals with several singular ends

Lemma

Improper convergence is independent of finite truncations and split points

Theorem

Linearity of convergent improper integrals

Theorem

Cauchy criterion for improper integrals

Definition

Absolute and conditional convergence of improper integrals

Theorem

A nonnegative improper integral converges iff its truncated integrals are bounded

Theorem

Absolute convergence implies improper convergence

Corollary

Tails of a convergent improper integral tend to zero

Theorem

Comparison tests for improper integrals

Corollary

Limit comparison for positive improper integrals

Corollary

Integral test as an equivalence with an improper integral

Lemma

Truncated integrals of rational powers

Theorem

The improper p-test for rational exponents

Definition

Cauchy principal values at a finite singularity and on the real line

Theorem

Separate improper convergence implies convergence of the principal value

Theorem

Dirichlet's test for improper integrals

Corollary

Abel's test for improper integrals

Theorem

A Dirichlet-type transfer criterion for divergence

Theorem

Uniform oscillatory tail mass forces failure of absolute convergence

Theorem

Monotone change of variable for Riemann-integrable functions

Theorem

Change of variable in an improper integral

Theorem

Frullani's formula with its proper integral factor

Remark

Conventions and proved scope for improper integrals

Sources