I'm working on a problem which involves two harmonic functions and defined in a square.

( , everywhere.)

The boundary conditions are along and along .

Along the functions satisfy the Cauchy Riemann equations

I know (via some dimensional analysis from the Navier Stokes equations) that because of the condition at these functionsmustsatisfy the Cauchy Riemann equations everywhere, and can show this with an asymtotic expansion from :

But, from the C-R at ,

and so on..

So that

p

But I'm struggling for a more formal proof... Could anyone point me in the right direction?

Thanks for your help!!