If $u(x,y)$ and $v(x,y)$ are harmonic on a domain D, show that $f(z) = (U_y – V_x) + i(U_x + V_y)$ is analytic in D.

Question

I am struggling on how to approach the following question:

If $u(x,y)$ and $v(x,y)$ are harmonic on a domain D, show that $f(z) = (U_y – V_x) + i(U_x + V_y)$ is analytic in D.

Any help would be appreciated.

Answer 1

Write $f(z) = g(x,y) + ih(x,y)$, where $g = U_y - V_x$ and $h = U_x + V_y$. To show $f$ analytic we can just show that $g,h$ satisfy the Cauchy-Riemann equations.

Then $g_x = U_{yx} - V_{xx}$ and $h_y = U_{xy} + V_{yy}$. Now, $h_y - g_x = U_{xy} + V_{yy} - U_{yx} + V_{xx}$, which is zero because $U_{xy}=U_{yx}$ and because $\Delta V = 0$. Thus $h_y = g_x$.

The other equation to show is $g_y = -h_x$, and proceeds similarly; see if you can do that one yourself.