# Prove if $u(x,y)$ is harmonic, then $u_x - iu_y$ is holomorphic

I can show the Cauchy-Riemann equations hold for $u_x - iu_y$, but Cauchy-Riemann does not imply holomorphic. If $-iu_y$ is a harmonic conjugate of $u_x$, then $u_x - iu_y$ is holomorphic, but I don't know how to show that two functions are harmonic conjugates without them being parts of a holomorphic function (which we have to prove).

• What else do you need in addition to Cauchy-Riemann (at your current stage)? As I recall you just need Cauchy-Riemann and a minor regularity hypothesis (that the derivatives are continuous, or something similar) to get holomorphicity.
– Ian
Commented Mar 24, 2015 at 17:34

## 1 Answer

Cauchy-Riemann (+ a small technical condition, depending on the presentation you're following) equations are necessary and sufficient to be holomorphic.

• The small technical condition which you can look up in your book will go through since $u(x,y)$ is harmonic (so has 2 continuous derivatives). Commented Mar 24, 2015 at 17:37