1550251515-Classical_Complex_Analysis__Gonzalez_

438 Chapter 7

Examples:

1. J 0 ez sin z dz = 0 for any simple closed curve C with graph in the
   complex plane, since f(z) = ez sinz is an entire function, i.e., analytic
   in C.

2. To evaluate the area of the ellipse C: z =a cost+ ibsint, 0::::; t 5 271",

we have

A= ~Jzdz= ~ f

2

1r(acost-ibsint)(-asint+ibcost)dt

2z 2z Jo

a

= ab1r

Corollan:y 7.8 Suppose that f = u +iv is conjugate analytic in a simply

or multiply connected region R and that fx = Ux + ivx is continuous in R.

If C is a simple closed contour homotopic to a point in R, then

j f(z)dz=O

a

Proof Since f is conjugate analytic in R, we have

fz = %[(ux +Vy)+ i(vx - uy)] = 0

so that Ux = -Vy and Vx = Uy in. R. Hence all partials. are continuous in
the same region. We have

j f(z)dz= j(udx+vdy)+i j(vdx-udy)

a a a

and on applying Green's theorem to each integral on the right, we obtain

/ f(z)dz= jj(vx-uy)dxdy-ijj(ux+vy)dxdy=O

C D D

where, as before, D denotes the region enclosed by C.

Example

Exercises 7 .1

j sinzdz = O

a

1. Evaluate the following integrals.