Alphabeta MathHow 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.
Conventions and proved scope for improper integrals
Statement
Every singular end is tested separately, and changing a finite split point does not change convergence or value. A finite limit is required: divergence to an infinite extended value is still divergence. Absolute and conditional convergence inherit the same piecewise convention.
A Cauchy principal value is a coupled symmetric limit and can exist without the corresponding improper integral. The -test on this page is proved for rational exponents only. Frullani's formula deliberately retains the proper factor ; no logarithm identity, Lebesgue-integrability statement, or arbitrary-real-exponent extension is claimed.
Depends on
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- Improper integrals with several singular ends
- Absolute and conditional convergence of improper integrals
- Cauchy principal values at a finite singularity and on the real line
- The improper $p$-test for rational exponents
- Frullani's formula with its proper integral factor
Used by
Dependency tree · two levels
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Sources
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)