1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 441

12. In Green's formula (7.9-2) let P = -GFy and Q = GFx, where Fis of

class c<^2 >(D) and G is of class C(l)(D), to show that

ff G\7^2 Fdxdy+ ff \7F·\7Gdxdy= f G~~ ds

D D C

dF / dn denoting the real directional derivative in the direction of the

exterior normal at points on the curve C, 'VF the gradient of F, and

\7^2 F its Laplacian. In particular, deduce that

ff \7^2 Fdxdy = f ~~ ds

D C

and that fc (dF/dn) ds = 0 whenever Fis harmonic in D.

13. Let f E C(l)(A), A open. If C is a simple closed contour around the

point z = ( x, y) E A such that C* C A, and <J" is the area bounded

by C*, show that

~f = lim _j_ r f(z) dz

uz ,,. ...... o 20" } 0

14. Prove that in complex notation Green's theorem takes the form

f M(z,z)dz + N(z,z)dz = 2i ff (Mz - Nz)dxdy

C D

7.10 The Cauchy-Goursat Theorem

As already noted, this is the same as Theorem 7.11 with the assumption of

the continuity off' dropped. Several proofs have been given of the Cauchy-

Goursat theorem. The following two lemmas concern with special cases of
the main theorem, while the next two lead to a proof of the theorem as
presented by W. F. Eberlein [11].

Lemma 7.1 (Goursat). Let f be analytic in a region G containing the

rectangle

R = {(x,y): a~ x ~ b,c ~ y ~ d}

Then

f !(z)dz=O

8R