1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 437

Since f is analytic in D, we have Ux = Vy and Uy = -vx. Hence each of

the double integrals on the right of (7.9-4) vanishes, and we get

J J(z)dz = 0

a

Corollary 7 .6 If f is analytic in R, f^1 is continuous in R, R is simply

connected, and the graph of the simple closed contour C is contained in

R, then

J f(z) dz= 0

a

Proof It follows at once from Theorem 7.11 since for R simply connected

C is always deformable to a point in R.

Note From (6.7-9), namely,

we have

2ifz = -(vx + uy) + i(ux - Vy)

and so, as anticipated in Section 6.10, (7.9-4) can also be written in the form

· J f(z) dz= 2i J J f:zdxdz (7.9-5)

a D

Of course, fz = 0 if f is analytic on D, and Cauchy's theorem follows as

before.
Clearly, if fa f dz= O, then La f dz= 0 also. In addition, the integral
is still zero if the contour is described a whole number of times in either
direction.

Corollary 7.7 If in formula (7.9-5) we take f(z) = z and let A= area
of D, we get

or

J z dz = 2i J J dx dy = 2iA

a D

A=~ jzdz

2i

a

(Mancill [19]) (7.9-6)