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one of my homework problems is the following:

When can a polynomial $p(x,y)$ be expressed as $p(x,y)=f(x+iy)+g(x-iy)$, for $f,g$ holomorphic?

I'm a bit lost after thinking about this question for a long while, as it seems rather vague. I would guess that it would only be possible if the Laurent series of $f(x+iy)$ and $g(x-iy)$ cancel each other out after a certain term, but I don't think this is a very satisfying answer. Would you have any thoughts as to how I might go about this problem? Thanks a bunch in advance.

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Have you tried Cauchy-Riemann equations? I'm trying myself, but I always get confused with partial derivatives. : - ) I think you might get necessary conditions at least. – k.stm Oct 3 '12 at 7:19
Thanks for the tip! I think saying that $p$ is harmonic is more like what the question is looking for. – MathStudent1208 Oct 3 '12 at 17:42
up vote 1 down vote accepted

Hint: harmonic functions. The fact that a harmonic function is the real part of an analytic function may be useful.

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So we must be summing the real parts of the holomorphic functions $f(x+iy)$ and $g(x-iy)$, and since a sum of harmonic functions is harmonic, $p(x,y)$ must be harmonic? – MathStudent1208 Oct 3 '12 at 17:18
$f(x+iy)$ is a holomorphic function of $z = x+iy$, and therefore is harmonic. $g(x-iy)$ is the complex conjugate of a holomorphic function, and therefore it is also harmonic. – Robert Israel Oct 3 '12 at 18:29
Got it, thanks! – MathStudent1208 Oct 3 '12 at 19:10

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