1550251515-Classical_Complex_Analysis__Gonzalez_

468 Chapter^7

Fig. 7.18

where BM P and P NB are the two arcs into which the contour el (suppos-

edly described once in the positive direction) is decomposed by the points

Band P. Then

so that

j f(z) dz - J f(z) dz= J f(z) dz - J f(z) dz

'Yl 'Y2 BMP PNB

= J f(z)dz=I

c+ 1

b1l r f(z) dz= (-y^2 J r f(z) dz+ I

lzo lzo

Hence if 'Yl and 'Y 2 are not homotopic in G with the same endpoints, the

integrals along 'Yl and '}' 2 may differ by a constant I f:. 0 called the module

of periodicity of the integral. Alternatively, this situation may occur when

the closed c.ontour /1 - 12 is not homotopic to a point in G.

We note that the contour e 1 ~ould be described m times as a part of 'Yl

and n times as a part of 12 ( m, n nonzero integers). In that case we have

and it follows that

'Y1 =AB+ me{ +BMP +PQ

'Y2 =AB+ net+ BNP+ PQ

b1) r f(z) dz= (1'2) r f(z) dz+ kl

lzo lzo

(k=m-n) (7.16-2)