CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-11

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$1 / x^{2}$ has no finite Cauchy principal value at zero

Example

Symmetry does not rescue the nonnegative singularity $1 / x^{2}$ : $PV ⁣ \int_{- 1}^{1} \frac{d x}{x^{2}}$ does not exist as a finite real number.

Facts & Assumptions

Given: The function $x^{- 2}$ away from zero.

[L1]

Principal value uses the sum of the two symmetric truncations (Cauchy principal values at a finite singularity and on the real line).

[L2]

The rational-power formula evaluates each proper truncation (Truncated integrals of rational powers).

Verification

technique · computation

1.1

For $0 < ε < 1$ , symmetry and [L2] give [L2] $\int_{- 1}^{- ε} \frac{d x}{x^{2}} + \int_{ε}^{1} \frac{d x}{x^{2}} = 2 (\frac{1}{ε} - 1) .$

2.1

This tends to $+ \infty$ as $ε ↓ 0$ , not to a finite real. Therefore the principal value in [L1] diverges.

L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)

1/x21/x2 has no finite Cauchy principal value at zero