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The study of the Dirac system and second-order elliptic equations withcomplex-valued coefficients on the plane leads to bicomplex Vekua equations. Tothe difference of complex pseudoanalytic (generalized analytic) functions thetheory of bicomplex functions has not been developed. Such basic facts as thesimilarity principle or the Liouville theorem in general are no longeravailable due to the presence of zero divisors in the algebra of bicomplexnumbers. We develop a theory of bicomplex pseudoanalytic formal powersanalogous to the developed by Bers and obtain Cauchy's integral formula in thebicomplex setting. In the classical complex situation this formula was obtainedunder the assumption that the involved Cauchy kernel is global, a restrictivecondition taking into account possible applications, especially when theequation itself is not defined on the whole plane. We show that the Cauchyintegral formula remains valid with a Cauchy kernel from a wider class calledhere the reproducing Cauchy kernels. We give a complete characterization ofthis class. To our best knowledge these results are new even for complex Vekuaequations. We establish that reproducing Cauchy kernels can be used to obtain afull set of negative formal powers for the corresponding bicomplex Vekuaequation and present an algorithm for their construction. Bicomplex Vekuaequations of a special form called main Vekua equations are closely related toSchr\"odinger equations with complex-valued potentials. We use this relation toestablish connections between the reproducing Cauchy kernels and thefundamental solutions for the Schr\"odinger operators which allow one toconstruct the Cauchy kernel when the fundamental solution is known and viceversa as well as to construct the fundamental solutions for the Darbouxtransformed Schr\"odinger operators.

Fundamentals of bicomplex pseudoanalytic function theory: Cauchy  integral formulas, negative formal powers and Schr\"odinger equations with  complex coefficients