1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

96 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS

Points of nonanalyticity for a function are called singular points. They
are important for certain applications in physics and engineering.
Our definition of the derivative for complex functions is formally the same
as for real functions and is the natural extension from real va~iables to complex
variables. The basic differentiation formulas a.re identical to those for real func-
tions, and we obtain the same rules for differentiating powers, sums, products,
quotients, and compositions of functions. We can easily establish the proof of
the differentiation formulas by using the limit theorems.
Suppose that f and g a.re differentiable. From Equation (3-1) and the tech-
nique exhibited in the solution to Example 3.1, we can establish the following
rules, which are virtually identical to those for real-valued functions.

:z C = 0, where C is a constant and (3-4)

:z z" = nz"-^1 , where n is a positive integer. (3-5)

d

dz[Cf(z)]=Cf'(z), (3-6)

d

dz [f (z) + 9 (z)] = f' (z) + 91 (z), (3-7)

! (J (z) g (z)] = f (z) g' (z) + g (z) f^1 (z), (3-8)

df(z) g(z)f'(z)- f(z)g'(z) providedthatg(z)fO, and

dz g (z) = [g (z)]^2

d

dz f (g (z)) = f' (g (z)) g' (z).

(3-9)

(3-10)

Important particular cases of Equations (3-9) and (3-10), respectively, are

d 1 -n

for z f 0 and n a positive integer, and

dz z" = zn+l'

:z [! (z)]" = n [f (z)]"-^1 f' (z), n a positive integer.

(3-11)

(3-12)

• EXAMPLE 3.3 If we use Equation (3- 12 ) with f (z) = z^2 + i2z + 3 and
  f' (z) = 2z + 2i, then we get

:z (z^2 +i2z + 3)

4

= 8 (z^2 + i2z+ 3)

3

(z+i).

The proofs of the rules given in Equations (3-4) through (3-10) depend on the

validity of extending theorems for real functions to their complex companions.

Equation (3-8), for example, relies on Theorem 3.1.