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

{{errata page
| chapter = Frequency Domain Design
| page = 340
| line = 3-9
| contributor = D. MacMynowski, A. Asimakapoulos
| date = 22 Nov 2010
| rev = 2.10c
| version = Third printing
| text = 
In the computation of the integral <math>I_3</math>, the analysis of the value of the contour integral is incorrect.  The corrected text is shown below in blue:
<blockquote>
Next we consider the integral
<math>I_3</math>.  For this purpose we split the contour into three parts <math>X_+</math>,
<math>\gamma</math> and <math>X_{-}</math>, as indicated in Figure 11.16. We
can then write the integral as
<center><amsmath>
  I_3 = \int_{X_+}\log{S(s)} \, ds+\int_\gamma\log{S(s)} \, ds+\int_{X-}\log{S(s)} \, ds.
</amsmath></center>
The contour <math>\gamma</math> is a small circle with radius <math>r</math> around the pole
<math>p_k</math>. The magnitude of the integrand is of the order <math>\log{r}</math>, and
the length of the path is <math>2\pi r</math>. The integral thus goes to zero as
the radius <math>r</math> goes to zero.
<font color=blue>
Since <amsmath>S(s)\approx k/(s-p_k)</amsmath> close to the pole, the argument of
<math>S(s)</math> decreases by <math>2\pi</math> as the contour encircles the
pole. On the
contours <math>X_+</math> and <math>X_{-}</math> we therefore have
<center><amsmath>
  |S_{X_+}|= |S_{X_-}|,\qquad \arg S_{X_-}=\arg S_{X_+} -2\pi.
</amsmath></center>
Hence
<center><amsmath>
   \log(S_{X_+})-\log(S_{X_-})=2\pi i,
</amsmath></center>
and we get
<center><amsmath>
  \int_{X_+}\!\!\log{S(s)}\, ds + \int_{X_-}\!\log{S(s)}\, ds = 
  2\pi\ i\ {\mathrm Re}\, p_k.
</amsmath></center>
Repeating the argument for all poles <math>p_k</math> in the right half plane,
</font>
letting the small circles go to zero and the large circle go to
infinity gives
<center><amsmath>
  I_1+I_2+I_3=-2i\int_0^{R}\log{|S(i\omega)|}d\omega+i\ \sum_k2\pi\ {\mathrm Re}\, p_k=0.
</amsmath></center>
Since complex poles appear as complex conjugate pairs, 
<amsmath>\textstyle \sum_k {\mathrm Re}\, p_k= \sum_k p_k</amsmath>, which gives Bode's
formula (11.19).
</blockquote>
}}