Treffer: Hypercomplex monogenic and areolar monogenic functions

Let us consider a partial differential equation of higher order with real constant coefficients of the form \[ P(D_x,D_y) u=\left(\sum^n_{k=0} \alpha_k D_x^{n-k} D_y^k\right)u=0,\;\alpha_n=1. \] To this equation one can associate an associative and commutative algebra of order \(n\) over the real field with the basis \(\{1,g,g^2, \dots,g^{n-1}\}\), where \(g\) satisfies \(\sum^n_{k=0} \alpha_kg^k=0\). The authors define hypercomplex functions and explain what does it mean that such a function is monogenic in a region \(D\subset \mathbb{R}^2\) or areolar monogenic in \(D\). Some very special examples are discussed.