1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 491

f(n)(z) = ~ J J(() d(

27ri ((-z)n+l

a+

(7.21-2)

These formulas give a representation in terms of integrals of the successive

derivatives of an analytic function at a point. We note that if f is analytic

at z then it is also analytic in some neighborhood of z, so by taking a

simple closed contour around z with graph contained in that neighborhood,

a representation of the successive derivatives by formulas (7.21-2) is always

possible.

The results of this section lead to the following theorem.

Theorem 7 .30 If a function f is analytic in a region R, then f has

derivatives of all orders at every point of R. Each of these derivatives is

also an analytic function in R, and also a continuous function in that region.

Remarks The preceding property may be stated by saying that if a com-
plex function has a first derivative at each point of a region R, then the
function has derivatives of all orders everywhere in R. This property is
not shared by real functions. For instance, J(x) = x^513 has a derivative
f' ( x) = 5fsx^213 at every point of the real line, but it fails to have a second
derivative at x = 0. Formula (7.21-1) also holds for a multiply connected
region, with G now denoting the total boundary of the region.

Corollary 7.15 In a region where f = u+iv is analytic, the real functions

u(x, y) and v(x, y) have partial derivatives of all orders with respect to x

and y, and all those partials are continuous functions in the same region.

Proof The existence of f' (z) implies, by Theorem 6.5,

J'(z) = Ux + ivx =Vy - iuy

the existence of f" ( z) implies

J"(z) = Uxx + ivxx = Vyx - iuyx = Vxy - iuxy

= - Uyy - ZVyy

and so on. The existence of f"(z) implies the continuity of J'(z), and so the

continuity of its components Ux = Vy, Vx = -Uy. Similarly, the existence of

f"'(z) implies the continuity of f"(z), hence the continuity of the partial

derivatives Uxx, Vxx, Uxy, Vxy, and so on; and, similarly, for the higher-order

partial derivatives of u and v. Formula (7.21-1) and, in particular, (7.21-2)

may be used for the evaluation of certain integrals.

Example To evaluate J 0 sinzdz/(z-7r)^4 where C: z = 4eit 0::::; t::::; 27r.

Here f(z) = sinz is analytic in the whole plane, and so is analytic on and