106 CHAPTER 3 8 ANALYTI C ANO HARMONIC FUNCTIONS
- EXAMPLE 3.7 At the beginning of this section (Equation (3-13)) we defined
the function f (z) = u (x, y) +iv (x, y) = x^3 - 3xy^2 + i (3x^2 y - y^3 ). Show that
this function is differentiable for all z, and find its derivative.
Solution We compute Ux (x, y) = v 11 (x, y) = 3x^2 -3y^2 and u 11 (x, y) = -6xy =
- v., (x, y), so the Cauchy- Riemann Equations (3-16) are satisfied. Moreover,