Knowlpedia

Definition

Cauchy principal value

A symmetric limiting prescription for certain singular or non-absolutely convergent integrals.

Definition

For a function gg singular at t=0t=0, its Cauchy principal value is

p.v.⁡ ⁣∫Rg(t) dt=lim⁡ε↓0∫∣t∣>εg(t) dt,\operatorname{p.v.}\!\int_{\mathbb R}g(t)\,dt =\lim_{\varepsilon\downarrow0} \int_{|t|>\varepsilon}g(t)\,dt,

provided the limit exists, with any additional truncation at infinity stated explicitly. The symmetric deletion around the singularity permits cancellation that an ordinary improper integral may not have.

Distributional interpretation

The prescription defines the p.v.⁡(1/t)\operatorname{p.v.}(1/t) by

⟨p.v.⁡1t,φ⟩=lim⁡ε↓0∫∣t∣>εφ(t)t dt.\left\langle\operatorname{p.v.}\frac1t,\varphi\right\rangle =\lim_{\varepsilon\downarrow0}\int_{|t|>\varepsilon} \frac{\varphi(t)}{t}\,dt.

It is the kernel underlying the .

Warning

Principal value convergence is weaker than absolute convergence and depends on the prescribed symmetric truncation. It must not be silently replaced by a .

References
  1. Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Publisher record.