1550251515-Classical_Complex_Analysis__Gonzalez_

552

14. Suppose that f(z) = L:::o anzn, JzJ < 1, and that

Jf(z)J:::; 1-1JzJ2

Prove that

Jani <^1 / 2 (n + 2)e

Chapter 8

15. Let f(z) = L:::'=o anzn for Jzl < R, and C: z = reit, 0 :::; t :::; 211',

r constant (0 :::; r < R). Prove Parseval's identity

1 1211" 00.

• Jf(reit)J2 dt = L Jan J2r2n
  211' 0 • n=O

16. (a) Apply Parseval's identity to the function f(z) = 1/(1-z) to show

that

_..!._ f 211" -~-d_t~~~ - ~1~

211' } 0 1 - 2r cost + r^2 1 - r^2

(O:::;r<l)

(b) Apply Parseval's identity to the function f ( z) = ( zn -1) / ( z -1) =

1 + z + · · · + zn-l, z f= 1, f(l) = n, to show that

1

2

11" ( sin

1

/ 2 nt )

2

_

. 1/ di - 21l'n
0 sm 2 t
17. With the same notation as in problem 13, suppose that Jf(z)J:::; M(r)
for JzJ = r < R, to prove that
00
L JanJ2r2n:::; M(r)2
n=O

8.6 Taylor Series for Nonanalytic Functions of Class C^00

For complex functions f = u+iv with components u and v of class cn+i in

a neighborhood of a point we derive a complex form of Taylor's formula. As

usual, the notation u, v E cn+l means that u and v are continuous and that

all their partial derivatives of order :::; n + 1 exist and are continuous in a

certain region (in this case some neighborhood of a given point). If u and v

are of class C^00 (i.e., if u and v are continuous and have continuous partial

derivatives of every order in that neighborhood), and certain additional

conditions on u, v and their partial derivatives are satisfied, then an infinite

Taylor series expansion of f(z) is valid (15].