Question

Find the principal value of the integral \(\int_{-\infty}^{\infty} x \sin x d x /\left(x^{2}-x_{0}^{2}\right)\) and evaluate $$ I=\int_{-\infty}^{\infty} \frac{x \sin x}{\left(x-x_{0} \pm i \epsilon\right)\left(x+x_{0} \pm i \epsilon\right)} d x $$ for the four possible choices of signs.

Short answer

The four integrals all evaluate to zero based on the Plemelj formula.

Step-by-step solution

Analyzing the Integrals

These four integrals are of the form \(\int_{-\infty}^{\infty} \frac{x \sin x}{(x−x_{0} \pm i \epsilon)(x+x_{0}\pm i \epsilon)}dx\). The values for the principal value of these integral forms need to be calculated.

Applying Plemelj Formula

There are four principal value integrals to be evaluated, using Plemelj formula the integral could be expressed in terms of the Cauchy principal value of the integral of a function f(x). The Plemelj formula is \(P.V. \int_{-\infty}^{\infty} \frac{f(x)}{x \pm i\epsilon} dx = \pm \frac{i}{2} f(0)\). Given that \(f(x) = \frac{x \sin x}{(x^{2}-x_{0}^{2})}\). We apply it to express the original integral form making sure to conduct this operation four times for all variants of the integrals given in the problem.

Calculation of the integrals

Now we apply Plemelj formula to the integrals. Using \(f(0)\) as \(\frac{x_{0}}{x_{0}^{2} - x_{0}^{2}} \sin(0) = 0\) for the denominators \(x-x_{0} \pm i\epsilon\) and \(x+x_{0}\pm i\epsilon\) we will have \(P.V. \int_{-\infty}^{\infty} \frac{x \sin x}{(x-x_{0} \pm i\epsilon)(x+x_{0} \pm i\epsilon)} dx = \pm \frac{i}{2} * 0\). After all the calculations, we notice that for all four conditions of our integral, the result is still zero.

Key concepts

Plemelj Formula

The Plemelj formula plays a crucial role in the field of complex integration, particularly when dealing with integrals that have singularities. It is a mathematical expression that allows us to resolve the ambiguity associated with integrals that are not defined in the conventional sense because of a singularity on the real line.

At its core, the Plemelj formula helps us to calculate the Cauchy principal value of an integral around a singularity by adding an infinitesimal imaginary part, denoted as \(i \epsilon\). In essence, this formula states that for a function \(f(x)\), the principal value integral of \(f(x)\) over a singularity at zero can be expressed as
\[P.V. \int_{-\infty}^{\infty} \frac{f(x)}{x \pm i\epsilon} dx = \pm \frac{i}{2} f(0)\]
By applying this to the given scenario where \(f(x)\) had a singularity, we observed that \(f(0)\) evaluates to zero due to the function's behavior at that point. Thus, by using the Plemelj formula, we could determine that the principal value of the integral regardless of the sign of \(\epsilon\) is also zero.

Cauchy Principal Value

When we talk about the Cauchy principal value, we refer to a technique devised to assign a value to certain improper integrals that would otherwise be undefined due to a singularity in the integrand or limits of integration that extend to infinity. It is especially useful in complex analysis, where singularities are common.

The principal value is computed by symmetrically excluding a small region around the singularity from the domain of integration and then taking the limit as the size of that excluded region goes to zero. For our integral, the principal part is denoted as \(\text{P.V.} \int_{-\infty}^{\infty} \), which suggests that we are dealing with the integration over the whole real line except at the singularity points. As demonstrated in the solution, even though the original integral cannot be evaluated in the standard sense due to the singularity at \(\text{x} = \pm \text{x}_0\), the Cauchy principal value provides a method to compute it, resulting in a definite value, in this case, zero.

Complex Integration

Complex integration extends the concept of integration into the complex plane, which brings about unique challenges and powerful tools such as the Cauchy principal value and the Plemelj formula. This form of integration is particularly powerful when evaluating integrals involving complex-valued functions over contours in the complex plane.

The integral provided in the exercise, \( \int_{-\infty}^{\infty} \frac{x \sin x}{(x^{2}-x_{0}^{2})} dx \) is an example of an integral that can be approached using complex integration techniques. By considering the integral along contours in the complex plane and avoiding singularities by incorporating an infinitesimal imaginary part \( \epsilon \), the original problematic integral can be transformed into a tractable form. This demonstrates how complex integration not only addresses singularities but also allows us to understand and compute integrals that would otherwise be intractable using real analysis alone.