Page 35 - Calculus with Complex Numbers

Ffgure 3.3

Proof Suppose f @) is differentiable at .x then we have
f( f(x + h) - flx) ?
x + h) - flx) = h h --> 0 x f (.x) = 0

Corollal'y flx) = 1/.x is not differentiable at .x = 0.

Observe thatthe converse of Theorem 1 is false. A counterexample is flx) = 1.x I
which is continuous but not differentiable at .x = 0.
For a complex function of a complex variable z, we deline differentiability and
continuity of flz) exactly as we have done for real functions of a real variable. The
familiar functions all have their familiar derivatives, and the familiar combination
rules are all valid. There is also a further constraint in the form of the Cauchy-
Riemann equations to which we devote the next section.

3.2 C auchy-Riem ann equations
Suppose we have a complex valued function w = flz) of the complex variable
z, and suppose we write w = u + i t? z = .x + iy then we can express u t?
as functions of .x y and corlsider their partial derivatives bu/bx élu/él y, é) v/bx
é) v/by. For example, if w = zl then
u + iv = w = .:2 = Lx + ïy)2 = .x2 + zj-xy - y2

which gives in this case
1: 2 2
= . Y - # y