1550251515-Classical_Complex_Analysis__Gonzalez_

164 Chapter3

Fig. 3.23

Other alternative definitions of a simply connected set will be given later.

3.16 The Winding Number of a Curve

Let "(: z = z( t), a :::; t :::; f3, be a (closed) curve, and let zo be a point of
the plane not on the curve, i.e., such that z 0 -:/:-z(t) for all t E [a,(3]. For
each t denote by Rt the ray with initial point at z 0 and passing through
z( t) (Fig. 3.23), i.e.,

Rt= {z: z = zo + (z(t)-z 0 )..\, 0:::; ,\ < +oo}

Ast varies from a to (3, the point z(t) describes the graph of"(, and the

ray Rt rotates about the fixed point z 0 until it reaches its initial position,
ROI = R13. Hence the ray makes a whole number of rotations about z 0 •
This integer is called the winding number or index of 'Y with respect to the
point zo (or about zo), and will be denoted by !1 7 (z 0 ). Other notations
found in the literature are n('Y,z 0 ), v('Y,z 0 ), and W("(,zo). The winding
number is regarded as positive if Rt rotates (ultimately) about z 0 in the

counterclockwise direction, as negative if Rt rotates in the clockwise direc-

tion, and as zero if Rt reverses its direction, returning to its initial position
without completing a rotation about z 0 (Fig. 3.24).
The foregoing is an intuitive approach to the concept of winding number
of a curve. To arrive at a rigorous definition, we note first that since

Fig. 3.24