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).

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    $\begingroup$ 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. $\endgroup$ – Ian Mar 24 '15 at 17:34

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

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    $\begingroup$ 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). $\endgroup$ – Batman Mar 24 '15 at 17:37

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