1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 467

converges, and we get

or

(7.15-4)

The same result is obtained if a is supposed to be negative, since cos 2ax

is an even function of a.

A more general method for evaluating real improper integrals is

discussed in detail in Section 9.11.

7.16 PRIMITIVES OF AN ANALYTIC FUNCTION IN A

MULTIPLY CONNECTED REGION

Let f be an analytic function defined on a multiply connected region G. If

j f(z)dz = 0

c

for every closed contour C with graph in G (or if the integral of f is

independent of the path in G), then Theorem 7.8 applies and the integral

z

F(z) ~bl j f(()d( (7.16-1)

zo

with z 0 , z E G, defines a single-valued analytic function F(z) on G such

that F'(z) = f(z).

But if there is at least a closed contour C 1 , with graph in G, such that

j'f(z)dz =I-/= 0

c+ 1

then the integral in (7.16-1) will have at least two different values at z, so
the function F(z) will be multiple-valued in G. In fact, suppose that the
path 'YI joining z 0 to z is taken to be (Fig. 7.18)

'Yi= AB+ BMP + PQ

while the path ')' 2 is chosen as

')' 2 =AB+ BNP+ PQ