1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.2 • THE CAUCHY-RIEMANN EQUATIONS 103

Note some of the important implications of this theorem.

• If f is differentiable at zo, then the Cauchy- Riemann Equations (3-16) will
  be satisfied at zo, and we can use either Equation (3-14) or (3-15) to evaluate
  f' (zo).


• Taking the contrapositive, if Equations (3-16) are not satisfied at zo, then
  we know automatically that f is not differentiable at zo.


• Even if Equations (3-16) are satisfied at zo, we cannot necessarily conclude
  that f is differentiable at zo.

We now illustrate each of these points.

• EXAMPLE 3.4 We know that f (z) = z^2 is differentiable and that f' (z) =
  2z. We also have

f (z) = z^2 = (x +iy)^2 = (x^2 - y^2 ) + i (2xy) = u (x,y) +iv (x,y).

It is easy to verify that Equations (3-16) are indeed satisfied:

u,. (x, y) = 2x =Vy (x, y) and u 11 (x, y) = -2y = - v,. (x, y).

Using Equations (3-14) and (3- 15 ), respectively, to compute f' (z) gives

f' (z) = u,.(x,y) +iv,. (x,y ) = 2x+i2y = 2z, and

f' (z) =Vy (x,y) - iuy (x,y) = 2x - i(- 2y) = 2x + i2y = 2z,

as expected.

• EXAMPLE 3.5 Show that f (z) = z is nowhere differentiable.

Solution We have f (z) = f (x + iy) = x - iy = u (x, y) +iv (x, y), where

u(x,y) = x and v(x,y) = - y. Thus, for any point (x,y), u,,(x,y) = 1 and

Vy (x, y) = -1. The Cauchy- Riemann equations are not satisfied at any point

z = (x, y), so we conclude that f is nowhere differentiable.