Proof of Analyticity of $w_{\varepsilon }(z)$ in the Cartesian Case

As a curiosity, it is interesting to point out that a first-order linear low-pass filter can be defined over a straight segment on the complex plane. In this way a filter in the Cartesian coordinates of the complex plane can be defined. Consider an analytic function $w(z)$ anywhere on the complex plane. Consider also a segment of length $2\varepsilon$ and a fixed direction given by a constant angle $\alpha$ with the real axis. Given an arbitrary position $z$ on the complex plane we define then two other points by

$$\begin{eqnarray*} z_{\oplus} & = & z+\varepsilon\,{\rm e}^{\mbox{\boldmath$\i... ... & = & z-\varepsilon\,{\rm e}^{\mbox{\boldmath$\imath$}\alpha}. \end{eqnarray*}$$

This defines a segment of length $2\varepsilon$ going from $z_{\ominus }$ to $z_{\oplus }$. Given any point $z$ such that this segment is contained within the analyticity domain of $w(z)$, we may now define a filtered function $w_{\varepsilon }(z)$ by

$$\begin{eqnarray*} w_{\varepsilon}(z) & = & \frac{1}{2\varepsilon} \int_{z_{\... ...left(z+\lambda\,{\rm e}^{\mbox{\boldmath$\imath$}\alpha}\right), \end{eqnarray*}$$

where $\lambda$ is a real parameter describing the segment, such that $-\varepsilon\leq\lambda\leq\varepsilon$ and

$$\begin{eqnarray*} z' & = & z+\lambda\,{\rm e}^{\mbox{\boldmath$\imath$}\alpha... ... dz' & = & \,{\rm e}^{\mbox{\boldmath$\imath$}\alpha}d\lambda. \end{eqnarray*}$$

Since $w(z)=u(x,y)+\mbox{\boldmath$\imath$}v(x,y)$ is analytic, $u(x,y)$ and $v(x,y)$ are continuous, differentiable and satisfy the Cauchy-Riemann conditions in Cartesian coordinates. We may write for $w_{\varepsilon }(z)$

$$\begin{eqnarray*} w_{\varepsilon}(z) & = & u_{\varepsilon}(x,y) + \mbox{\bo... ...ilon}d\lambda\, v[x+\lambda\cos(\alpha),y+\lambda\sin(\alpha)]. \end{eqnarray*}$$

It is now clear that $u_{\varepsilon}(x,y)$ and $v_{\varepsilon}(x,y)$ are also continuous and differentiable. If we now take the partial derivatives of these functions with respect to $x$ we get

$$\begin{eqnarray*} \frac{\partial u_{\varepsilon}}{\partial x}(x,y) & = & \fra... ...ial v}{\partial x}[x+\lambda\cos(\alpha),y+\lambda\sin(\alpha)]. \end{eqnarray*}$$

Using the Cauchy-Riemann conditions for $u(x,y)$ and $v(x,y)$ we get

$$\begin{eqnarray*} \frac{\partial u_{\varepsilon}}{\partial x}(x,y) & = & \fra... ...ial v}{\partial x}[x+\lambda\cos(\alpha),y+\lambda\sin(\alpha)]. \end{eqnarray*}$$

Taking now the partial derivatives of $u_{\varepsilon}(x,y)$ and $v_{\varepsilon}(x,y)$ with respect to $y$ we get

$$\begin{eqnarray*} \frac{\partial u_{\varepsilon}}{\partial y}(x,y) & = & \fra... ...ial v}{\partial y}[x+\lambda\cos(\alpha),y+\lambda\sin(\alpha)]. \end{eqnarray*}$$

Comparing this pair of equation with the previous ones we finally get

$$\begin{eqnarray*} \frac{\partial u_{\varepsilon}}{\partial x}(x,y) & = & \fra... ...) & = & -\, \frac{\partial v_{\varepsilon}}{\partial x}(x,y). \end{eqnarray*}$$

This establishes that $u_{\varepsilon}(x,y)$ and $v_{\varepsilon}(x,y)$ satisfy the Cauchy-Riemann conditions, and therefore that $w_{\varepsilon }(z)$ is analytic. Once $w_{\varepsilon }(z)$ is defined by the filter at all points of the domain of analyticity of $w(z)$ where the segment fits, and now that it has been proven analytic there, one can extend the definition of $w_{\varepsilon }(z)$ to the whole domain of analyticity of $w(z)$ by analytic continuation.