Nové integrální formule v hyperkomplexní analýze

Sikora, Martin

Title: New Integral Formulae in Hypercomplex Analysis Author: Mgr. Martin Sikora Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Vladimír Souček, DrSc., MÚ UK Supervisor's e-mail address: soucek@karlin.mff.cuni.cz Abstract: The Dirac equation for Clifford algebra-valued functions on the even-dimensional Minkowski space can be understood as a hyperbolic sys- tem of partial differential equations. We show how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula which expresses the value of a function in a chosen point as an integral over a compact cycle given by the intersection of the null cone with the upper sheet of the hyperboloid in the Minkowski space. We also treat the ultra-hyperbolic case where the Dirac equation gives the ultra-hyperbolic system of partial differential equations. An analogue of the second order Cauchy formula is proved for (n − 1)-vector-valued holo- morphic functions. It reconstructs values inside a bounded domain in the 2n-dimensional complex space by integrating over the characteristic boun- dary of the domain. 1

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