Questions tagged [cauchy-principal-value]

Suppose $f$ has a singularity at $c\in (a,b)$, and for each $\epsilon>0$, $f$ is integrable on $(a,c-\epsilon)$ and $(c+\epsilon,b)$. Then, the Cauchy principal value of $\int _a^b f(x) dx$ is defined as $$ \operatorname{pv\!\!}\int_a^b f(x) dx := \lim_{\varepsilon\to 0^+} \int _a^{c-\varepsilon} f(x)\,dx + \int _{c+\varepsilon}^b f(x)\,dx $$ The principal value can exist even if the integral does not. For instance, although $f(x)=1/x,x\neq 0$ is not integrable on $[-1,1]$ since neither sided integral converges, the principal value of the integral is zero by cancellation. If $f$ is improperly integrable on $[a,b]$ anyway, the prinicpal value agrees with the usual result. The principal value is defined similarly over an infinite range of integration: to assign a value to $\int_{\mathbb{R}} f(x)\,dx$, we take $$ \lim_{a\to +\infty} \int _{-a}^{a} f(x)\,dx $$ There are similar definitions for a function with finitely many singularities on $\mathbb R$.

See also the Wikipedia page on the Cauchy principle value.