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.