The problem at hand is this:
letting $u, v : \mathbb{R}^2 \rightarrow \mathbb{R}$ be two functions, assume $v$ is the harmonic conjugate of $u$ on $\mathbb{R}^2$. Show that $uv$ is the imaginary part of an entire function and conclude that $uv$ is harmonic.
This is my take on the question, that i'm not very sure of:
$u$ and $v$ are harmonic conjugates, therefore a function $g = u + iv$ is holomorphic and we have the Cauchy Reimann equations (1):
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$ $$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
let $uv$ be a the imaginary part of a function $f$, and $a$ be its real part. So we have $f = a + iuv$ which should be holomorphic. Therefore, it should satisfy the Cauchy Reimann equations (2):
$$\frac{\partial a}{\partial x} = \frac{\partial (uv)}{\partial y} = v\frac{\partial u}{\partial y} + u \frac{\partial v}{\partial y}=v\frac{\partial u}{\partial y} + u\frac{\partial u}{\partial x}$$ $$\frac{\partial a}{\partial y} = -\frac{\partial(uv)}{\partial x} = - v\frac{\partial u}{\partial x} -u\frac{\partial v}{\partial x} = -v\frac{\partial u}{\partial x} + u\frac{\partial u}{\partial y}$$
Where I have replaced equations (1) in (2) to get all the derivatives is terms of u.
To see if this is satisfied I computed the laplacian of $a$, and making all the developments I got $\frac{\partial^2 a}{\partial x^2} + \frac{\partial^2 a}{\partial y^2} = 0$.
This is the part I'm not sure of, I concluded that $a$ is therefore harmonic and that the Cauchy reimann equations are indeed satisfied, therefore $f$ is indeed holomorphic over the whole plane (entire), thus $uv$ is harmonic.
Is this work remotely okay? Any help or comment is much appreciated.
