Mathematics of Physics and Engineering

(^200) Functions of a Complex Variable
To complete the proof of (4.2.11), it is therefore enough to show that
lira d> /(C) - /()
P-O 2m JCo <-z
-dC = 0,
and the argument is very similar to the proof of Theorem 4.2.2; see the
discussion following equation (4.2.9). Indeed, the continuity of / implies
that, for every e > 0, we can find p > 0 so that \f(z) — /(C)| < £ as long as
|C ~ z\ < P- Using inequality (4.2.8) and keeping in mind that LQP = 2-Kp
and |C — z\ = p when £ G Cp, we find:
2m Jc
/(C) ~ f(z) ,r
—7 "C < e, (4.2.13)
which completes the proof. •
Similar to Remark 4.2, if the domain G is bounded and has a piece-wise
smooth boundary CQ, and the function / is analytic in G and is continuous
in the closure of G, then, for every z £ G, we have
'
(
"-55jL^
(4
'
2

EXERCISE 4.2.18. (a)B Fill in the details leading to inequality (4.2.13).

(b)c What is the value of the integral on the right-hand side of (4-2.11) if

the point z does not belong to the closure of the domain enclosed by C.

Hint: 0 by the Integral Theorem of Cauchy.

EXERCISE 4.2.19. A (a) Use the Integral Formula of Cauchy to prove

the mean-value property of the analytic functions: if f is analytic in

a domain G (not necessarily simply connected), ZQ is a point in G, and

{z : \z — zo\ < p} is a closed disk lying completely inside G, then

1 /"27r

/(Z°) = 2W /(0+Pe")d- ( (^4) - (^2) - (^15) )
Hint: Use (4-2.14) in the closed disk; ( = z 0 + pezt. (b) In (4-2.15, can you
replace the average over the circle with the average over the disk? Hint: yes.
An almost direct consequence of representation (4.2.11) is that an an-
alytic function is differentiable infinitely many times. More precisely, if
/ = fiz) is analytic in a domain G (not necessarily simply connected), the