1550251515-Classical_Complex_Analysis__Gonzalez_

436 Chapter 7

Fig. 7.5

a proof of the theorem can be given that does not make use of the con-

tinuity of the derivative. This so-called Cauchy-Goursat theorem will be

discussed in Section 7.10.

Proof (Cauchy [9], 1846: Riemann [32], 1851). This proof depends on

Green's theorem for the plane (15], namely: Let P(x,y) and Q(x,y) be

two real-valued functions defined on a closed domain D bounded by a

rectifiable Jordan curve C, and suppose that

1. P(x, y) and Q(x, y) are continuous on D.


2. Py and Qx exist and are continuous on D.


3. C is described once in the positive (counterclockwise) direction. Then

j P dx + Q dy = j j ( Q x -: Py) dx dy t (7.9-2)

C D

Now, writi~g (7.8-14) for the contour C, we have

j f(z)dz= j(udx-vdy)+ij(vdx+udy) (7.9-3)

c c c

and applying (7.9-2) to each of the real line integrals on the right, we obtain

j f(z)dz = jj(-vx - uy)dxdy + i j j(ux -vy)dxdy (7.9-4)

C D D

tThis theorem, which is a two-dimensional analog of the fundamental theorem
of integral calculus, although attributed to Green, was known to Lagrange and
Gauss. Several versions of the theorem are found in the literature, depending on
the restrictions imposed on D, C, and the partial derivatives Py, Q,,, (see [2], [35],
and [39]). S. Bochner in [6] establishes a Green's theorem by replacing assumption
(2) by the total differentiability of P and Q together with the continuity of
Q,, - Py on D.