1550251515-Classical_Complex_Analysis__Gonzalez_

310 Chapter6

y

(^0) x
Fig. 6.1
The concept of the derivative at a point may be modified by considering

the function f defined on sets other than open sets (or, f restricted to such

sets). For example, we could let the domain of definition off be any subset
B of the complex plane having at least an accumulation point zo E B, and
define a restricted derivative at z 0 by the formula

f1(zo) = ti.z->0 lim

zo+.C..zEB

f(zo + ~z) - f(zo)

~z

if the limit exists, with the condition that z 0 + ~z varies on B only. This
concept is useful, for instance, when B is a closed region and z 0 a boundary
point of B, zo being accessible from B (Fig. 6.1). In such a case the
derivative at z 0 may exist only in that restricted sense. The point z 0 + ~z
could be restricted further to an angular region contained in B and with
vertex at z 0 • This leads to the so-called angular derivative, which has been
considered by some authors (see, e.g., Caratheodory [21], vol. II, p. 32).
In Section 6.9 we discuss the special case where B is a regular arc z =
z(t), a :::; t ::::; /3 (Fig. 6.2). The resulting limit is called the directional
derivative off at z 0 • For a nonopen set EC <C and a function f: E--? <C,
analyticity on E may be defined either in a global or in a local manner,
as follows:

y

0

~Liz

s/zo

x

Fig. 6.2