Cauchy Principal Value in R

Subjects: Real Analysis
Links: Improper Integrals in R

The Cauchy Principal value is a is a method for assigning values to certain improper integrals which would otherwise be undefined.

For a singularity at a finite number b

We can also define over singularities. Let $a < b < c$ , where $b$ is a difficult point at which the behaviour of the $f$ is such that

\int_{a}^{b} f (x) d x = \pm \infty

for any $a < b$ , and

\int_{b}^{c} f (x) d x = \mp \infty

Then we can define the Cauchy Principle value for a singularity at a finite number $b$ , as

p.v. \int_{a}^{c} f (x) d x = lim_{ε \to 0^{+}} [\int_{a}^{b - ε} f (x) d x + \int_{b + ε}^{c} f (x)]

For a singularity at $\infty$

The Cauchy principle Value at infinity is defined as:

p.v. \int_{R} f = lim_{x \to \infty} \int_{- x}^{x} f

where

\int_{- \infty}^{0} f (x) d x = \pm \infty

and

\int_{0}^{\infty} f (x) d x = \mp \infty $ $ I n t h e s p e c i a l c a s e t h a t : $ f : R \to R $, i f $ \int_{R} f $ e x i s t s, t h e n : $ $ p.v. \int_{R} f = \int_{R} f

In some cases it is necessary to deal simultaneously with singularities both at a finite number $b$ and at infinity. This is usually done by a limit of the form

lim_{η \to 0^{+}} lim_{ε \to 0^{+}} [\int_{b - \frac{1}{η}}^{b - ε} f (x) d x + \int_{b + ε}^{b + \frac{1}{η}} f (x) d x] .

In those cases where the integral may be split into two independent, finite limits,

lim_{ε \to 0^{+}} | \int_{a}^{b - ε} f (x) d x | < \infty

and

lim_{η \to 0^{+}} | \int_{b + η}^{c} f (x) d x | < \infty

According to the my Complex Analysis by Paez:
Given a function $f : R {x_{1}, \dots, x_{n}} \to R$ such that $x_{k} < x_{k + 1}$ for $k \in {1, \dots, n - 1}$ and $f$ is not bounded for any neighbourhood around $x_{k}$ , and integrable over any interval $[c, d] \subseteq R ∖ {x_{1}, \dots, x_{n}}$ , we define the Cauchy principal value of the integral $\int_{R} f$ as

p.v. \int_{- \infty}^{\infty} f (x) d x = lim_{α \to \infty} (\int_{- α}^{x_{1} - 1 / α} f (x) d x + \sum_{k = 1}^{n} \int_{x_{k} + 1 / α}^{x_{k + 1} - 1 / α} f (x) d x + \int_{x_{k} + 1 / α}^{α} f (x) d x)

which can be the same from the definition from wikipedia but don't know how.

Calculating the Cauchy principal value can be useful, since if $f$ is even and integrable over any interval in $R$ we get that

\int_{0}^{\infty} f (x) d x = lim_{α \to \infty} \int_{0}^{α} f (x) d x = \frac{1}{2} lim_{α \to \infty} \int_{- α}^{α} f (x) d x = \frac{1}{2} p.v. \int_{- \infty}^{\infty} f (x) d x

For a singularity at a finite number b

For a singularity at ∞

For a singularity at $\infty$