Complex Analysis - Day 2
October 18, 2007
Topics: Logarithmic derivative, winding number, argument principle, counting zeros and poles, Rouche’s Theorem.
Problems:
J03#3, S06#5, J05#2, S05#4, J04#1, J90#1, J91#5, S98#5.
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Meromorphic A function f is meromorphic in a domain D if is is analytic
in D except for poles.
Winding Number Let Γ be a simple closed contour that does not pass
through 0. The winding number of Γ is the number of times the contour winds
around the origin, which is positive if it goes around counterclockwise, and negative if it goes around clockwise. If C is a positively oriented simple closed
contour, f is meromorphic in the domain enclosed by C and nonzero on C, and
Γ = f (C), then the winding number of Γ can be calculated from the change in
the argument of f (z) as f goes around C:
winding number of Γ =
1
∆C arg f (z).
2π
Argument Principle Suppose f is meromorphic in the domain interior to a
positively oriented simple closed contour C, and f is analytic and non-zero on
C. Let Z be the number of zeros of f inside C, including multiplicity, and P
be the number of poles. Then
Z
1
f 0 (z)
1
dz =
∆C arg f (z) = Z − P.
2πi C f (z)
2π
Rouche’s Theorem Suppose that f and g are analytic inside and on a simple
closed contour C, and |f (z)| > |g(z)| at each point on C. Then f and f + g
have the same number of zeros, counting multiplicities, inside C.
1