Errata: Contour integral in derivation of Bode's integral formula is analyzed incorrectly

Location: page 340, line 3-9

In the computation of the integral $\text{[math]}$ , the analysis of the value of the contour integral is incorrect. The corrected text is shown below in blue:

Next we consider the integral $\text{[math]}$ . For this purpose we split the contour into three parts $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ , as indicated in Figure 11.16. We can then write the integral as

The contour $\text{[math]}$ is a small circle with radius $\text{[math]}$ around the pole $\text{[math]}$ . The magnitude of the integrand is of the order $\text{[math]}$ , and the length of the path is $\text{[math]}$ . The integral thus goes to zero as the radius $\text{[math]}$ goes to zero. Since close to the pole, the argument of $\text{[math]}$ decreases by $\text{[math]}$ as the contour encircles the pole. On the contours $\text{[math]}$ and $\text{[math]}$ we therefore have

Hence

and we get

Repeating the argument for all poles $\text{[math]}$ in the right half plane, letting the small circles go to zero and the large circle go to infinity gives

Since complex poles appear as complex conjugate pairs, , which gives Bode's formula (11.19).