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

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Location: page 340, line 3-9

In the computation of the integral $I 3$ , the analysis of the value of the contour integral is incorrect. The corrected text is shown below in blue:

Next we consider the integral $I 3$ . For this purpose we split the contour into three parts $X +$ , $γ$ and $X -$ , as indicated in Figure 11.16. We can then write the integral as
The contour $γ$ is a small circle with radius $r$ around the pole $p k$ . The magnitude of the integrand is of the order $log r$ , and the length of the path is $2π r$ . The integral thus goes to zero as the radius $r$ goes to zero. Since close to the pole, the argument of $S (s)$ decreases by $2π$ as the contour encircles the pole. On the contours $X +$ and $X -$ we therefore have

Hence

and we get

Repeating the argument for all poles $p k$ 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).