Prove that if u and v have continuous mixed partials and satisfy the Cauchy-Riemann5 equat...

Prove that if u and v have continuous mixed partials and satisfy the Cauchy-Riemann5 equation

then both u and v are harmonic.

5 Augustin-Louis Cauchy (1789−1857), a Frenchman, contributed to such areas as geometry, analysis, number theory, and mechanics. His textbooks on analysis presented calculus in substantially the form that it is taught in present-day introductory courses. Bernhard Riemann (1826−1866) contributed to mathematics in general through his analytic perspective. In particular, he used analysis to elucidate problems in number theory and geometry. The process of integration presented in introductory calculus texts, called Riemann integration in his honor, was but one small part of his work.