# 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 , the analysis of the value of the contour integral is incorrect. The corrected text is shown below in blue:

Next we consider the integral . For this purpose we split the contour into three parts , and , as indicated in Figure 11.16. We can then write the integral as

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

Hence

and we get

Repeating the argument for all poles 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).

(Contributed by D. MacMynowski, A. Asimakapoulos, 22 Nov 2010)