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Absolute and conditional convergence of improper integrals
Statement
An improper integral of is absolutely convergent when the corresponding improper integral of converges. It is conditionally convergent when the integral of converges but the integral of does not.
For an integral with several singular ends, absolute convergence means absolute convergence on every separately defined one-ended piece. Conditional convergence means convergence of every piece and failure of absolute convergence on at least one piece. Thus the terminology never permits cancellation between distinct singular ends.
Depends on
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- Improper integrals with several singular ends
- If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$
- Absolute value in an ordered field
Used by
- The sine integral is improperly Riemann integrable and not Lebesgue integrable Counterexample
- ∫_-∞^∞(1+x²)⁻¹ dx converges absolutely Example
- A step function whose improper integral is the alternating harmonic series Example
- Conventions and proved scope for improper integrals Remark
- A Dirichlet-type transfer criterion for divergence Theorem
- Absolute convergence implies improper convergence Theorem
- Dirichlet's test for improper integrals Theorem
- Uniform oscillatory tail mass forces failure of absolute convergence Theorem
Dependency tree · two levels
22 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)