332 CHAPTER 8 • RESIDUE T HEORYvFigure 8. 12 The image curve f (C2 (O)) under f (z) = z^2 + z.
radian measure off ( C) around the origin. Therefore, when we divide both sides
of Equation (8-36) by 271'i, its right side becomes an integer (by Theorem 8.8)
that must count the number of times f ( C) winds around the origin.
• EXAMPLE 8. 25 The image of the circle C 2 (0) under f (z) = z^2 + z is the
curve {(x,y) = (4cos2t+2cost,4sin2t+ 2sint): 0 < t < 211'} shown in Figure
8.12. Note that the image curve f (C2 (0)) winds twice around the origin. We
check this by computing 2 ~; fct<o)^1 ;cWdz = 2 ~, .fct(o) ~~$!dz. The residues of
the integrand are at 0 and - 1. Thus,
-^1 1 --2z+ld z = .R es [ 2z + l --o] + R es [ 2--z+l - 1 ]
271'i ct(o) z^2 + z z^2 + z ' z^2 + z'
= 1+1=2.
Finally, we note that if g (z) = f (z) - a, then g' (z) = f (z) , and thus we
can generalize what we've just said to compute how many times the curve f ( C)
winds around the point a. Theorem 8.9 summarizes our discussion.