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Improper integrals with several singular ends
Statement
If has a possible singularity at , define only when the two one-sided improper integrals converge separately. Cancellation between divergent sides is not allowed.
Likewise, for any finite split point , define 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
Used by
- Absolute and conditional convergence of improper integrals Definition
- |x-c|^-1/2 has a convergent improper integral across an interior singularity Example
- ∫_-∞^∞(1+x²)⁻¹ dx converges absolutely Example
- Convergence range of x⁻ᵖ(1+x)^-q on (0,∞) for rational exponents Example
- Improper convergence is independent of finite truncations and split points Lemma
- The improper integral of e^-x² over ℝ is finite and positive Lemma
- Conventions and proved scope for improper integrals Remark
- Change of variable in an improper integral Theorem
- Frullani's formula with its proper integral factor Theorem
- Linearity of convergent improper integrals Theorem
- Separate improper convergence implies convergence of the principal value Theorem
Dependency tree · two levels
13 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)