1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

Other notations for the derivative are w', Df(z), and

Examples

1. Let f(z) = z^2 , G = <C. In this case we have

dw

""JZ•

J'(z) = lim (z + ~z)

2

• z

2

= lim (2z + ~z) = 2z

~z->O ~z ~t->O

for all z E <C.

2. Let J(z) = lzl^2 = zz, G = <C. In this example we find that

f(z + ~z)-J(z)

~z

= (z -'-~-'-~~ + ~z)(z + ~z) -zz

~z -

=z+z ~z +~z

309

If we write ~z = rei^8 , we get ~z = re-ie and ~z/~z e-zie,

so that

f(z + ~;;-f(z) = z + ze-2;e + re-ie (6.1-3)

Suppose that we let ~z -+ 0 by keeping the argument () fixed and letting
r -+ 0. Then the right-hand side of (6.1-3) will tend to z + ze-zie, which
depends on () except at the point z = 0. Thus the given function does
not possess a derivative in the complex plane, except at the origin, so that
in this case G 1 = {O} and J'(O) = 0. The given function is said to be
monogenic at z = 0. In Section 6.2 we give an example of a function, in
fact, a continuous function, that does not have a derivative anywhere in
the complex plane.

Definitions 6.2 A function f having a derivative at a point z 0 E A (A

open) is said to be monogenic or differentiable at z 0.

If f has a derivative at z 0 , as well as at every point of some neighborhood

of z 0 , it is said to be analytic or holomorphic at z 0 • If f is analytic at every

point of A, then f is analytic or holomorphic in A. A function that is

analytic in <C is said to be entire or integral. t

Examples w = z^3 and w = ez are entire functions.
Of course, if a function is not defined at zo, or, if it fails to have a
derivative there, it is not analytic at the point.

Example w = l/z is not analytic at z = 0. However, it is analytic in any
open set that does not contain z = 0.

tThe term holomorphic, from the Greek 'o>.o, =entire and μop<pfJ =form, means

of a form analogous to that of the entire functions.