1550251515-Classical_Complex_Analysis__Gonzalez_

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Hence (8.6-4) can be written in the form

n 1

f(z + D.z) = f(z) + L kl (fzD.z + f-zD."i)(k) +Rn

k=l

Chapter 8

(8.6-5)

This is Taylor formula for complex functions that are of class cn+i in a
neighborhood of z.

Theorem 8.13 With the same notation as in Theorem 8.12, suppose now
that u and v are of class C^00 in Nr(z) and that Rn~ 0 as n ~ oo. Then
we have the Taylor series representation

00 1

f(z + D.z) = f(z) + L kl (fzD.z + fzD.z)(k)

k=l

(^00) i d-
= f(z) + L kl (fz + fz d; )(k) dzk
k=l
= f(z) + f: t! Jik)(z) dzk
k=l
(8.6-6)

by letting; dz= ldzleiA and denoting by Jik)(z) the kth complex rectilinear

directional derivative of f at z in the direction specified by >..

The condition Rn ~ 0 as n ~ oo certainly holds in the following cases:

1. The partial derivatives of u and v are in absolute value uniformly
   bounded in Nr(z).


2. The functions u and v together with their partial derivatives of all
   orders are nonnegative in Nr(z).

3. f is analytic or conjugate analytic in Nr(z).

Proof Clearly, under the assumption u,v E C^00 in Nr(z), Rn~ 0 is both

a necessary and sufficient condition for the validity of (8.6-6). Thus it
remains to consider the cases under which Rn ~ 0.
In case (1) let the partial derivatives of u be in absolute value uniformly
bounded by M 1 , while those of v be in absolute value uniformly bounded
by M 2 • Then we have
M 2n+IM
IR1,nl ~ (n +)l (ID.xi+ ID.v[)n+l ~ (n + l)~ ID.zln+l

and similarly,

2 n+1M

IR 2,n I ~ (n+l)!^2 ID.zln+l