10.5: Cauchy Principal Value

First an example to motivate defining the principal value of an integral. We’ll actually compute the integral in the next section.

Example \(\PageIndex{1}\)

Let

\[I = \int_{0}^{\infty} \dfrac{\sin (x)}{x} \ dx. \nonumber \]

This integral is not absolutely convergent, but it is conditionally convergent. Formally, of course, we mean

\[I = \lim_{R \to \infty} \int_{0}^{R} \dfrac{\sin (x)}{x} \ dx. \nonumber \]

We can proceed as in Example 10.3.3. First note that \(\sin (x) /x\) is even, so

\[I = \dfrac{1}{2} \int_{-\infty}^{\infty} \dfrac{\sin (x)}{x} \ dx. \nonumber \]

Next, to avoid the problem that \(\sin (z)\) goes to infinity in both the upper and lower half-planes we replace the integrand by \(\dfrac{e^{ix}}{x}\).

We’ve changed the problem to computing

\[\tilde{I} = \int_{-\infty}^{\infty} \dfrac{e^{ix}}{x}\ dx. \nonumber \]

The problems with this integral are caused by the pole at 0. The biggest problem is that the integral doesn’t converge! The other problem is that when we try to use our usual strategy of choosing a closed contour we can’t use one that includes \(z = 0\) on the real axis. This is our motivation for defining principal value. We will come back to this example below.

Definition: Cauchy Principal Value

Suppose we have a function \(f(x)\) that is continuous on the real line except at the point \(x_1\), then we define the Cauchy principal value as

\[\text{p.v.} \int_{-\infty}^{\infty} f(x) \ dx = \lim_{R \to \infty, r_1 \to 0} \int_{-R}^{x_1 - r_1} f(x)\ dx + \int_{x_1 + r_1}^{R} f(x)\ dx. \nonumber \]

Provided the limit converges. You should notice that the intervals around \(x_1\) and around \(\infty\) are symmetric. Of course, if the integral

\[\int_{-\infty}^{\infty} f(x)\ dx \nonumber \]

converges, then so does the principal value and they give the same value. We can make the definition more flexible by including the following cases.

1. If \(f(x)\) is continuous on the entire real line then we define the principal value as
   \[\text{p.v. } \int_{-\infty}^{\infty} f(x) \ dx = \lim_{R \to \infty} \int_{-R}^{R} f(x)\ dx \nonumber \]
2. If we have multiple points of discontinuity, \(x_1 < x_2 < x_3 < \ ... < x_n\), then
   \[\text{p.v. } \int_{-\infty}^{\infty} f(x)\ dx = \lim \int_{-R}^{x_1 - r_1} f(x)\ dx + \int_{x_1 + r_1}^{x_2 - r_2} + \int_{x_2 + r_2}^{x_3 - r_3} + \ ... \int_{x_n + r_n}^{R} f(x)\ dx. \nonumber \]

Here the limit is taken as \(R \to \infty\) and each of the \(r_k \to 0\) (Figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Intervals of integration for principal value are symmetric around \(x_k\) and \(\infty\). (CC BY-NC; Ümit Kaya)

The next example shows that sometimes the principal value converges when the integral itself does not. The opposite is never true. That is, we have the following theorem.

Example \(\PageIndex{2}\)

If \(f(x)\) has discontinuities at \(x_1 < x_2 < \ ... < x_n\) and \(\int_{-\infty}^{\infty} f(x) \ dx\) converges then so does \(\text{p.v.} \int_{-\infty}^{\infty} f(x) \ dx\).

Solution

The proof amounts to understanding the definition of convergence of integrals as limits. The integral converges means that each of the limits

\[\begin{array} {r} {\lim_{R_1 \to \infty , a_1 \to 0} \int_{-R_1}^{x_1 - a_1} f(x)\ dx} \\ {\lim_{b_1 \to 0, a_2 \to 0} \int_{x_1 + b_1}^{x_2 - a_2} f(x) \ dx} \\ {...} \\ {\lim_{R_2 \to \infty , b_n \to 0} \int_{x_n + b_n}^{R_2} f(x) \ dx.} \end{array} \nonumber \]

converges. There is no symmetry requirement, i.e. \(R_1\) and \(R_2\) are completely independent, as are \(a_1\) and \(b_1\) etc.

The principal value converges means

\[\lim \int_{-R}^{x_1 - r_1} + \int_{x_1 + r_1}^{x_2 - r_2} + \int_{x_2 + r_2}^{x_3 - r_3} + \ ... \int_{x_n + r_n}^{R} f(x)\ dx \nonumber \]

converges. Here the limit is taken over all the parameter \(R \to \infty, r_k \to 0\). This limit has symmetry, e.g. we replaced both \(a_1\) and \(b_1\) in Equation 10.6.9 by \(r_1\) etc. Certainly if the limits in Equation 10.6.9 converge then so do the limits in Equation 10.6.10. \(\text{QED}\)

Example \(\PageIndex{3}\)

Consider both

\[\int_{-\infty}^{\infty} \dfrac{1}{x} \ dx \ \ \ \text{and} \ \ \ \text{p.v.} \int_{-\infty}^{\infty} \dfrac{1}{x} \ dx. \nonumber \]

The first integral diverges since

\[\int_{-R_1}^{-r_1} \dfrac{1}{x} \ dx + \int_{r_2}^{R_2} \dfrac{1}{x} \ dx = \text{ln} (r_1) - \text{ln} (R_1) + \text{ln} (R_2) - \text{ln} (r_2). \nonumber \]

This clearly diverges as \(R_1, R_2 \to \infty\) and \(r_1, r_2 \to 0\).

On the other hand the symmetric integral

\[\int_{-R}^{-r} \dfrac{1}{x} \ dx + \int_{r}^{R} \dfrac{1}{x}\ dx = \text{ln} (r) - \text{ln} (R) + \text{ln} (R) - \text{ln} (r) = 0. \nonumber \]

This clearly converges to 0.

We will see that the principal value occurs naturally when we integrate on semicircles around points. We prepare for this in the next section.