8.7 • THE ARGUMENT PRINClPLE AND RoUCHE'S'THEOREM 329t Corollary 8.1 Suppose that f is analytic in the simply connected domain D.
Let C be a simple closed positively oriented contour in D such that for z E C,
J (z) f= 0. Then
1 r f I (z)
21ri Jc f (z) dz= ZJ,where Zt is the number of zeros off that lie inside C.
Remark 8.3 Certain feedback control systems in engineering must be stable.A test for stability involves the function G (z) = 1 + F (z), where Fis a rational
function. If G does not have any zeros in the region {z: Re(z);::: O}, then the
system is stable. We determine the number of zeros of G by writing F (z) = ¢f;j,
where P and Qare polynomials with no common zero. Then G (z) = Q(itJM,
and we can check for the zeros of Q (z)+P (z) by using Theorem 8.8. We select a
value R so that G (z) =F 0 for {z: lzl > R} and then integrate along t he contour
consisting of the right half of the circle CR (0) and the line segment between iR
and -iR. This method is known as the Nyquist stability criterion. •
Why do we label Theorem 8.8 as the argument principle? T he answer lies
with a fascinating application known as the winding number. Recall that a
branch of the logarithm function, log"' is defined by
log"z = lnlzl +iarg"'z = lnr + i¢>,
where z = re;q, =F 0 and a < <P ~ a+ 271'. Loosely speaking, suppose that for
some branch of the logarithm, the composite function log"(! (z)) were analytic
in a simply connected domain D containing the contour C. T his would imply
that log"(! (z)) is an antiderivative of the function^1 ;(w for all z E D. T he-
orems 6.9 and 8.8 would then tell us that, as z winds around the curve C, the
quantity logo:(f (z)) = lnl/(z)I + iargaf (z) would change by 21T'i(ZJ - P1).
Since 2ni (Z 1 - Pt) is purely imaginary, this result tells us that arga f (z) would