1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 495

small enough), and similarly, F+(z 0 ) denotes the limit of F(z) as z -t z 0

from the right along a nontangential path. Formula (1) is called the first

Plemelj formula, and (2) is called the second Plemelj formula [25].

7 .22 MORE!li¥S THEOREM

The following theorem, due to G. Morera [23], is a sort of converse of the

Cauchy-Goursat fundamental theorem.

Theorem 7.31 If the function f is continuous in the region G, and if

J f(z)dz = 0 (7.22-1)

G

for every closed contour C with graph in G, then f is analytic in G.

Proof By Theorem 7.5 the function

F(z) = 1z f(()d(,

zo

zo, z E G, zo fixed

is single-valued in G, i.e., independent of the path, with graph in G, and

joining zo and z. By Theorem 7.8 the function F is analytic in G and

F'(z) = f(z). Then it follows that f, being the derivative of an analytic

function, is itself analytic in G.

Remark The part of the hypothesis in Theorem 7.31, requiring that

(7.22-1) holds for every closed contour C, may be weakened. In fact, it

suffices to assume that each zo E G b~ the center of a circle 'Y contained

in G such that (7.22-1) hold for every closed contour C with graph con-

tained in Int 1*. Then it will follow that f is analytic in the interior of

1*, and so analytic in G. Alternatively, we need only assume that (7.22-1)
holds whenever C is the boundary T of a triangle such that T U Int T C G
(Exercises 7.5 (1)).
For a generalization of Morera's theorem due to J. Wolff, see S. Saks
(35], p. 196. Another generalization will be found in S. P. Ponomarev (28],
pp. 360-363. See also Exercises 7.5 problem 2.

7.23 Cauchy's Inequality

Theorem 7 .32 Let f be analytic in a region G containing the circle

C: z - zo = reit, 0:::; t:::; 271". If lf(z)I :::; Mon C, then

lf(n)(zo)I :::; M:! (n = 0, 1, 2, ... ) (7.23-1)

r