If \(f\) is continuous on the interval \(b\lt x\le c\text{,}\) but discontinuous at \(b\text{,}\) then the improper integral of \(f\) over \([b,c]\) is defined by
provided the limit exists. Similarly, if \(f\) is continuous on the interval \(a \le x\lt b\text{,}\) but discontinuous at \(b\text{,}\) then the improper integral of \(f\) over \([a,b]\) is defined by
If \(f\) is continuous for all values of \(x\) in the interval \([a,c]\text{,}\) except at the value \(x=b\text{,}\) where \(a\lt b\lt c\text{,}\) then the Cauchy principal value of \(f\) over \([a,c]\) is defined by
In this section we show how to use residues to evaluate the Cauchy principal value of the integral of \(f\) over \((-\infty, \infty)\) when the integrand \(f\) has simple poles on the \(x\)-axis. We state our main results and then look at some examples before giving proofs.
Theorem8.5.2.
Let \(f(z) =\frac{P(z)}{Q(z)}\text{,}\) where \(P\) and \(Q\) are polynomials with real coefficients of degree \(m\) and \(n\text{,}\) respectively, and \(n \ge m+2\text{.}\) If \(Q\) has simple zeros at the points \(t_1, \, t_2, \, \ldots, \, t_{l}\) on the \(x\) axis, then
where \(z_1, \, z_2, \, \ldots, \, z_k\) are the poles of \(f\) that lie in the upper half-plane.
Theorem8.5.3.
Let \(P\) and \(Q\) be polynomials of degree \(m\) and \(n\text{,}\) respectively, where \(n \ge m+1\text{,}\) and let \(Q\) have simple zeros at the points \(t_1, \, t_2, \, \ldots, \, t_{l}\) on the \(x\)-axis. If \(\alpha\) is a positive real number and if \(f(z) =\frac{\exp(i\alpha z)P(z)}{Q(z)}\text{,}\) then
where \(z_1, \, z_2, \, \ldots ,z_k\) are the poles of \(f\) that lie in the upper half-plane.
Remark8.5.4.
The formulas in these theorems give the Cauchy principal value of the integral, which pays special attention to the manner in which any limits are taken. They are similar to those in Sections 8.3 and Section 8.4, except here we add one-half the value of each residue at the points \(t_1, \, t_2, \, \ldots, \, t_l\) on the \(x\)-axis.
Example8.5.5.
Evaluate P.V. \(\int_{-\infty}^{\infty}\frac{x}{x^3-8}\,dx\) by using complex analysis.
The integrand \(f(z) = \frac{\exp(iz)}{(z-1)(z^2+4)}\) has simple poles at the points \(t_1=1\) on the \(x\)-axis and \(z_1=2i\) in the upper half-plane. By Theorem 8.5.3,
Suppose that \(f\) has a simple pole at the point \(t_0\) on the \(x\)-axis. If \(C_{r}\) is the contour \(C_{r}:z=t_0+re^{i\theta}\text{,}\) for \(0 \le \theta \le \pi\text{,}\) then
Since \(f\) has only a finite number of poles, we can choose \(r\) small enough that the semicircles
\begin{equation*}
C_j : z = t_j+re^{i\theta}, \text{ for } 0 \le \theta \le \pi \text{ and } j=1,2,\ldots ,l
\end{equation*}
are disjoint and the poles \(z_1, \, z_2, \, \ldots, \, z_k\) of \(f\) in the upper half-plane lie above them, as shown in Figure 8.5.10.
Figure8.5.10.The poles \(t_1, \ t_2, \ \ldots, \ t_l\) of \(f\) that lie on the \(x\)- axis and the poles \(z_1, \ z_2, \ \ldots \ z_k\) that lie above the semicircles \(C_1, \ C_2, \ \ldots, \ C_{l}\)
Let \(R\) be large enough so that the poles of \(f\) in the upper half-plane lie under the semicircle \(C_R:z=Re^{i\theta}\text{,}\) for \(0 \le \theta \le \pi\text{,}\) and the poles of \(f\) on the \(x\)-axis lie in the interval \(-R \le x \le R\text{.}\) Let \(C\) be the simple closed positively oriented contour that consists of \(C_R\) and \(-C_1, \, -C_2, \, \ldots, \, -C_l\) and the segments of the real axis that lie between the semicircles shown in Figure 8.5.10. The residue theorem gives \(\int_Cf(z)\,dz = 2\pi i\sum\limits_{j=1}^k\mathrm{Res}[f,z_j]\text{,}\) which we rewrite as
where \(I_R\) is the portion of the interval \(-R \le x \le R\) that lies outside the intervals \((t_j-r,t_j+r)\) for \(j=1,2,\ldots,l\text{.}\) Using the same techniques that we used in Theorems 8.3.4 and Theorem 8.4.1 yields
If \(f\) is the function given in Theorem 8.5.2, then Equation (8.5.8) becomes Equation (8.5.1). If \(f\) is the function given in Theorem 8.5.3, then equating the real and imaginary parts of Equation (8.5.8) results in Equations (8.5.2) and (8.5.3), respectively, which completes the proof.
