1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 459

arbitrarily closely by a piecewise regular contour, namely, an inscribed

polygon.

To justify the definitions above, we shall prove anew the following

properties.

Theorem 7.16 The winding number Da(a), as defined by (7.13-1), is

and integer.

Proof Let C: z = z(t), o: St S ,8, z(o:) = z(,B), and consider the function

1 1t z'(t)

·Gt = - dt

() 27ri °' z(t)-a

This function is continuous on [a:, ,8] and differentiable on [a:, ,8] except at

the points tk of discontinuity of z'(t), if any. Hence fort 'f=. tk (o: St S ,8),

we have

Then

G'(t) = _1_ z'(t)

27ri z(t) - a

~ e-^2 1riG(t)[z(t) - a]= e-^2 11"iG(t)[z'(t) - 27riG'(t)(z(t) - a)]

dt

=0

(7.13-2)

except possibly at a finite number of points. Since e-^2 1riG(t)[z(t) - a] is

continuous on [a:, ,BJ, it must reduce to a constant }( on [a:, ,8], i.e.,

e-21riG(t)[z(t) -a]=}(

or

z(t)-a= J(e21riG(t) (7.13-3)

Clearly,}( 'f=. 0 since z(t) -a 'f=. 0. Letting t = ,8 in (7.13-3) we have

}( e21riG(f3) = z(,8) a = z( o:) a = }( e211"iG(a) = }(

since z(,B) = z( o:) and G( o:) = 0. Thus e^2 11"iG(f3) = 1, which shows that
G(,8) = n (an integer).

Theorem 7 .1 7 Let C be a piecewise regular closed contour and a r/. C*.

Then Dc(a), as defined by (7.13-1), is a continuous function of a. If Mis

a connected set not intersecting C*, then Da(a) is a constant on M, and
if M is unbounded this constant is zero~