6.5 • INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS 239We now state two important corollaries of Theorem 6.12.t Corollary 6 .2 If f is analytic in the domain D , then, for integers n <:: 0, all
derivatives f (n) (z) exist for z E D (and therefore are analytic in D).
Proof For ea.ch point zo in D, there exists a closed disk lz - zol ~ R that is
contained in D. We use the circle C = Cn (zo) = {z: lz -zol = R} in Theorem
6.12 to show that j(n) (zo) exists for all integers n <:: 0.
Remark 6 .3 This result is interesting, as it illustrates a big difference between
real and complex functions. A real function f can have the property that f'
exists everywhere in a domain D, but/" exists nowhere. Corollary 6.2 states
that if a complex function f has the property that f' exists everywhere in a
domain D, then, remarkably, all derivatives off exist in D. •
t Corollary 6.3 If u is a harmonic function at each point (x, y) in the domain
D, then all partial derivatives u.,, Uy, u.,,,, Uxy, and u 1111 exist and are harmonic
functions.
Proof For ea.ch point zo = (xo, Yo) in D there exists a disk Dn. (zo) that is
contained in D. In this disk, a conjugate harmonic function v exists, so the
function f (z) = u +iv is analytic. We use the Cauchy-Riemann equations
to get f' (z) = Ux +iv., = Vy - iu 11 , for z E D11. (zo). Since f' is analytic in
D11. (zo), the functions u., and u 11 are harmonic there. Again, we can use the
Cauchy- Riemann equations to obtain, for z E D11. (zo),
!"() Z = Uxx +iv.,.,=. Vyx - tUyx. = - u 1111 - ivyy. ·
Because f " is analytic in DR. ( zo), the functions u.,.,, u., 11 , and u 1111 are harmonic
there.
-------~EXERCISES FOR SECTION 6.5Recall that c: (zo) denotes the positively oriented circle {z : lz - zol = p}.
- Find fctco> (exp z + cosz) z -^1 dz.
- Find fc t<•> (z + 1)-^1 (z - 1)-^1 dz.
- Find fc t<•> (z + 1) -^1 (z - 1)-^2 dz.
- Find fctc 1 > (z^3 - lr' dz.
- Find fct(o) z-^4 sinz dz.
- Find fctco> (z cosz)-
1
dz.