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 anywhere on the complex plane. Consider also a segment of length and a fixed direction given by a constant angle with the real axis. Given an arbitrary position on the complex plane we define then two other points by
This defines a segment of length going from to . Given any point such that this segment is contained within the analyticity domain of , we may now define a filtered function by
where is a real parameter describing the segment, such that and
Since is analytic, and are continuous, differentiable and satisfy the Cauchy-Riemann conditions in Cartesian coordinates. We may write for
It is now clear that and are also continuous and differentiable. If we now take the partial derivatives of these functions with respect to we get
Using the Cauchy-Riemann conditions for and we get
Taking now the partial derivatives of and with respect to we get
Comparing this pair of equation with the previous ones we finally get
This establishes that and satisfy the Cauchy-Riemann conditions, and therefore that is analytic. Once is defined by the filter at all points of the domain of analyticity of where the segment fits, and now that it has been proven analytic there, one can extend the definition of to the whole domain of analyticity of by analytic continuation.