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Cauchy principal values at a finite singularity and on the real line
Statement
Let be properly Riemann integrable on compact subintervals of , where . Its Cauchy principal value at is provided this coupled limit is finite.
For a locally Riemann-integrable function on the real line, define again only for a finite limit. A principal value couples the two truncations; it does not assert that the two one-sided improper integrals converge separately.
Depends on
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
- Limits at $+\infty$ and $-\infty$, and infinite limits at a point
- The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$
- The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$
Used by
- 1/x² has no finite Cauchy principal value at zero Counterexample
- Standard semicircle, rectangle, keyhole, indentation, and sector contours Definition
- 1/x on [-1,1] has principal value 0 but no improper integral Example
- The whole-line principal value of sin x / x is pi, so the half-line integral is pi / 2 Example
- FALSE: existence of a Cauchy principal value forces improper convergence False statement
- Conventions and proved scope for improper integrals Remark
- This page keeps Cauchy principal values distinct from genuine improper convergence Remark
- Indented real-axis contours compute principal values with half-residue corrections Theorem
- Separate improper convergence implies convergence of the principal value Theorem
Dependency tree · two levels
27 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)