If the two real-valued fields $u(x,y)$ and $v(x,y)$ are harmonic conjugates, as explained here, then
- There exists $f(x + i y) = u(x,y) + i v(x,y)$.
- $u$ and $v$ are linked by the Cauchy-Riemann equations.
- $u$ and $v$ satisfy Laplace's equation $\nabla^2 u = 0$ and $\nabla^2 v = 0$.
As it says on the Wikipedia article, the operator transforming $u$ into $v$ is called the Hilbert transform, or alternatively a Bäcklund transform.
The Hilbert transform
\begin{equation}H(u)(x) = \frac{1}{\pi}\lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} \frac{1}{x-y}u(y) \, dy,\end{equation}
is defined for one-dimensional cases in the Cauchy principal value sense. My first question is that what is the proper generalisation of the Hilbert transform into 2D (and higher)?
As far as I know, the most commonly taken higher-dimensional generalisation of the Hilbert transform is the Riesz transform
\begin{equation} \begin{split} R_jf(x)&=\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\lim_{\epsilon\to 0}\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy\\ &=\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\lim_{\epsilon\to 0}\int_{\mathbf{R}^n\backslash B_\epsilon(x)}\frac{(y_j-x_j)f(y)}{|x-y|^{n+1}}\,dy, \end{split} \end{equation} where $j = 1, \ldots, n$ and $y_j$ is the $j$th component of $y$ in $\mathbf{R}^n$. Also $x \in \mathbf{R}^n$. In 2D, the Cauchy principal value here means that first we take the integral in the original domain where a disk of radius $\epsilon$ has been excised around $x$ and then $\epsilon$ is made to tend to 0.
My second (and main) question is: what is the explicit form of the integral transformation that takes $u(x,y)$ into $v(x,y)$ and vice versa? Is it the Riesz transform? Then again, the Riesz transform is the set $\lbrace R_j\rbrace$ of $n$ separate transforms. I only want one transform between $u$ and $v$.
For example, let us take $u(x,y) = \frac{1}{2} \ln{[(x-x_0)^2 + (y-y_0)^2]}$, as shown e.g. here. Solving through the Cauchy-Riemann system, it is easily shown that
$$f(z)=\frac12 \log((x-x_0)^2 + (y-y_0)^2)+i\arctan2(y-y_0,x - x_0)=\log(z-z_0),$$ where $z = x + iy$. For the Laplace equation condition to hold, $u$ should have a harmonic conjugate if and only if we focus on some simply connected subset of $R^2\setminus \lbrace (x_0,y_0) \rbrace$.
So, I would like to know the explicit integral transformation $H$ such that
$$\arctan2(y-y_0,x - x_0) = H \left[ \frac12 \log((x-x_0)^2 + (y-y_0)^2) \right].$$ Presumably $H$ is the Bäcklund/Hilbert/Riesz transform in two dimensions.
A related question: if we know that $(-\partial_y S, \partial_x S) = (A_1,A_2) \equiv \vec{A}$, where in general $\vec{A} \neq \nabla F$ for some scalar field $F$, how to write an explicit integral for $S$? I suppose when we have the special case of $\vec{A} = \nabla F$, then $S$ and $F$ are related through the Cauchy-Riemann equations, and then they should just be a Hilbert/Riesz/Bäcklund transform apart, relating to my questions above? What about the general case of $\vec{A} \neq \nabla F$?
Thanks!