Hi, can anyone help me prove the following.
Suppose that u(x,y) and v(x,y) are smooth functions satisfying the Cauchy-Riemann equations. Prove that v(x,y) is harmonic.
thank you
Yes, but the important question is, "Can you prove it?"
The Cauchy-Riemann equations are:
(1) $\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$
(2) $\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$.
Differentiate (1) with respect to y and (2) with respect to x.
Remember that, as long as the second derivatives are continuous,
$\displaystyle \frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}$.