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Improper convergence is independent of finite truncations and split points
Statement
Changing a finite lower endpoint of , a finite upper endpoint of , or a compact truncation beside a finite singular endpoint neither creates nor destroys improper convergence. The values change by the corresponding oriented proper integral.
Consequently, convergence and value in the definitions of an interior-singularity integral and a whole-line integral are independent of the chosen finite split point.
Facts & Assumptions
Given: Local Riemann integrability on every compact interval away from the stated singular ends.
Proper integrals are additive over adjacent intervals (For : is integrable on if and only if it is integrable on and on , and then ; with the oriented form for arbitrary ).
At an infinite or finite one-sided singular end, adding or subtracting a fixed finite constant preserves convergence and translates the limit by that constant; this follows directly from the same epsilon estimate in the defining limits (Improper integrals over unbounded intervals, Improper integrals at a finite singular endpoint).
Improper integrals require separate finite limits at each singular end (Improper integrals over unbounded intervals, Improper integrals at a finite singular endpoint, Improper integrals with several singular ends).
Proof
If , then for every , [L1, L2] The first term is fixed and finite. Subtracting it changes the absolute error from a proposed translated limit by exactly the same amount, so [L2] shows that either truncation limit exists exactly when the other does and that their values differ by . The other three one-ended orientations follow by the same identity with endpoints reversed.
Let be two split points on the whole line. Step 1.1 transfers the finite proper integral from the right tail to the left tail, so the two sums agree. The same calculation around an interior singularity changes only the nonsingular finite portion. Because [L3] continues to require both pieces separately, no cancellation of divergent pieces is introduced.
Depends on
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- Improper integrals with several singular ends
- For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then $\int_a^b f = \int_a^c f + \int_c^b f$; with the oriented form for arbitrary $a,b,c$
- The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$
Used by
- Integral test as an equivalence with an improper integral Corollary
- Tails of a convergent improper integral tend to zero Corollary
- A nonnegative improper integral converges iff its truncated integrals are bounded Theorem
- Absolute convergence implies improper convergence Theorem
- Comparison tests for improper integrals Theorem
- Separate improper convergence implies convergence of the principal value Theorem
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
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)