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Show that if \phi (x,y)\, is harmonic then \phi _{x}-i\phi _{y}\, is analytic.


If \phi (x,y)\, is harmonic then \phi _{{xx}}+\phi _{{yy}}=0\,.

\phi _{x}-i\phi _{y}\, is analytic iff the Cauchy-Riemann equations hold.

In this case the C-R equations are \phi _{{xx}}=-\phi _{{yy}}\, and \phi _{{xy}}=\phi _{{yx}}\,.

The first equation is true because \phi (x,y)\, is harmonic and the second equation is true from the usual properties of partial derivatives.

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