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The Question

Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic.

I'm trying to prove that this function is also harmonic when $v$ is harmonic conjugate for $u$. I know that when this is the case, the function $f(z) = u(x,y) + iv(x,y)$ is analytic, and thus, $u+v$ or $uv$ are also harmonic. But I am not sure how to deal with the powers. Please help!

Thank you

Jimmy Dur
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Cathy Kim
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1 Answers1

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Suppose $u$ is harmonic (and real valued) and $v$ is a harmonic conjugate of $v$. Then $u + iv$ is analytic. Thus $(u + iv)^3$ is analytic. Thus the real part of $(u + iv)^3$ is harmonic. Because the real part of $(u + iv)^3$ equals $u^3 - 3uv^2$, we conclude that $u^3 - 3uv^2$ is harmonic.

Sheldon Axler
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    Thank you!!! That really helps! I only thought about regular (a+b)^3 and didn't see that with i^2, it would make the second term negative. I did not see that! Thank you! – Cathy Kim Jan 26 '16 at 12:04