1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 469

If the connectivity of G is p > 2 and the contours C1, C 2 , ••• , Cp-l are

not null homotopic in G, then if we set

Ir= J f(z)dz (r = 1, ... ,p-1)

a+ r

we obtain, in a similar fashion,

("Yi) r f(z) dz= ("Y^2 ) r f(z) dz+ kiI1 + · · · + kp-1Ip-1

lzo lzo

(7.16-3)

where the kr are integers. Again, the constants Ir are called modules of
·periodicity (or simply periods) of the integral. Hence

F(z) = ("Y) r J(() d(

lzo

is in general multiple-valued, each value generating a single-valued branch

of F(z), and any two of those branches differing by some constant. As in

the proof of Theorem 7.8 we obtain

F'(z) = J(z)

for any branch of F, provided that z remains in a simply connected

subregion G 1 C G.

The preceding results are summarized in the following theorem,

Theorem 7.22 Let f be analytic in the multiply connected region G.

Then the integral

F(z) = ("Y) r J(() d(,

lzo

zo, z E G, "/ C G

defines in general a multiple-valued function F(z) which has single-valued

analytic branches on any simply connected subregion G1 C G, and

F'(z) = f(z),

F' denoting the derivative of any of those branches of F.

Example Consider the function defined by the integral

F(z) = ("Y) r d(

11 (

(7.16-4)

in the punctured plane G = C - {O} (a doubly connected region). The

integrand J(() = 1/( is analytic in G and the integral in (7.16-4) is inde-

pendent of the path as long as 'Y does not wind about the origin. However,