3.2 • THE CAUCHY-RIEMANN EQUATIONS 107u, v, Ux, uy, v.,, and Vy are continuous everywhere. By Theorem 3.4, f is
differentiable everywhere, and, from Equation (3-14),
f' (z) = u,, ( x, y) + iv,, ( x, y) = 3x^2 - 3y^2 + i6xy = 3 ( x^2 - y^2 + i2xy) = 3z^2.
Alternatively, from Equation (3-15),
!' (z) =Vy (x, y) -iuy (x, y)
= 3x^2 - 3y^2 - i (-6xy) = 3 (x^2 - y^2 + i2xy) = 3z^2 •
This result isn't surprising because (x + iy)^3 = x3 - 3xy^2 + i (3x^2 y - y3), and
so the function f is really our old friend f ( z) = z^3.
- EXAMPLE 3.8 Show that the function f (z) = e-Y cos x + ie-Y sin x is dif-
ferentiable for all .z and find its derivative.
Solution We first write u (x, y) = e-Y cosx and v (x, y) = e- Y sin x and then
compute the partial derivatives.
u.,(x,y)=vy(x,y)= - e-Ysinx and
Vx (x, y) = -u 11 (x, y) = e-Y cosx.
We note that u, v, Ux, Uv, Vx, and Vy are continuous functions and that the
Cauchy- Riemann equations hold for all values of (x,y). Hence, using Equation
(3-14), we write
f' (z) = f '(x + iy) = Ux (x, y) + ivx (x, y) = -e- 11 sinx + ie- 11 cosx.
The Cauchy- Riemann conditions are particularly useful in determining the
set of points for which a function f is differentiable.
- EXAMPLE 3.9 Show that the function f (z) = x^3 + 3xy^2 + i (y^3 + 3x^2 y) is
differentiable on the x-and y-axes but analytic nowhere.
S olutio n Recall (Definition 3.1) that when we say a function is analytic at
a point zo we mean that the function is differentiable not only at zo, but also
at every point in some c: neighborhood of zo. With this in mind, we proceed
to determine where the Cauchy-Riemann equations are satisfied. We write
u(x, y) = x^3 +3xy2 and v (x, y) = y^3 +3x^2 y and compute the partial derivatives:
Ux (x,y) = 3x^2 +3y^2 , v 11 (x,y) = 3x^2 +3y^2 , and
Uy (x,y) = 6xy, Vx (x, y) = 6xy.