On pseudo-abalytic matrix functions

Youvaraj, G. P.; Jain, Rajendra K.

doi:10.1080/17476939008814457

Cited by 2 publications

(1 citation statement)

References 3 publications

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“…As the term w = K8 _ 1 h(z) is such a solution we get for the solution to of system (3) the representation w(z,() = K8 9(z) + (c((,z))'!-(K8 h(z))' (20) and the following theorem holds. (19) and (20) hold. c) Conversely, for arbitrary holomorphic vectors g and h the expressions in (19) and (20)…”

Section: N) A:kafl_kc = (A-1kafl_k%)*mentioning

confidence: 97%

On the Representation of Generalized Analytic Vectors

Berglez¹

1995

Z. Anal. Anwend.

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This paper is concerned with the representation of generalized analytic vectors which are defined as solutions of certain elliptic systems of linear first-order partial differential equations in two complex variables. For the explicite representation of solutions we give integral operators of Vekua type as well as differential operators of Bauer type which map holomorphic vectors onto the set of solutions. For the existence of the differential operators we give a necessary and sufficient condition. We discuss properties of these operators and show how to construct the kernels of the integral operators using the differential operators. Finally we present examples for systems of differential equations for which all the solutions can be represented explicitely.

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Section: N) A:kafl_kc = (A-1kafl_k%)*mentioning

confidence: 97%

On the Representation of Generalized Analytic Vectors

Berglez¹

1995

Z. Anal. Anwend.

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Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

Campos

Kravchenko

2012

Complex Anal. Oper. Theory

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The study of the Dirac system and second-order elliptic equations with complexvalued coefficients on the plane naturally leads to bicomplex Vekua-type equations [8], [14], [6]. To the difference of complex pseudoanalytic (or generalized analytic) functions [3], [25] the theory of bicomplex pseudoanalytic functions has not been developed. Such basic facts as, e.g., the similarity principle or the Liouville theorem in general are no longer available due to the presence of zero divisors in the algebra of bicomplex numbers.In the present work we develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by L. Bers [3] and especially that of negative formal powers. Combining the approaches of L. Bers and I. N. Vekua with some additional ideas we obtain the Cauchy integral formula in the bicomplex setting. In the classical complex situation this formula was obtained under the assumption that the involved Cauchy kernel is global, a very restrictive condition taking into account possible practical applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with the Cauchy kernel from a wider class called here the reproducing Cauchy kernels. We give a complete characterization of this class. To our best knowledge these results are new even for complex Vekua equations. We establish that reproducing Cauchy kernels can be used to obtain a full set of negative formal powers for the corresponding bicomplex Vekua equation and present an algorithm which allows one their construction.Bicomplex Vekua equations of a special form called main Vekua equations are closely related to stationary Schrödinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrödinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa. Moreover, using these results we construct the fundamental solutions for the Darboux transformed Schrödinger operators.

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On pseudo-abalytic matrix functions

Cited by 2 publications

References 3 publications

On the Representation of Generalized Analytic Vectors

On the Representation of Generalized Analytic Vectors

Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

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