The objective of this paper is to contribute towards the quality investigation of equations of conservation for two dimensional incompressible flow patterns via a Complex Analysis approach.

Specifically, the authors will utilize the aforementioned equations expressed in their known from literature equivalent special formulation in terms of the stream function Y and its partial derivatives with respect to spatial coordinates.

In the sequel, by means of the Theory of Analytic Complex Functions, the authors will infer some quality information about the algebraic and geometrical properties for the family of functions which satisfy these equations for such flow patterns.

streamfunction,holomorphicfunction,analyticfunction,division ring, quaternion, Mittag-Leffler problem, holomorphic region.