Some properties of the Cauchy-type integral for the time-harmonic relativistic Dirac equation

✍ Scribed by Baruch Schneider; Michael Shapiro

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 176 KB
• Volume: 25
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.381

✦ Synopsis

Abstract

In this paper we study an analogue of the Cauchy‐type integral for the theory of time‐harmonic solutions of the relativistic Dirac equation in case of a piece‐wise Liapunov surface of integration and we prove the Sokhotski–Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given Hölder function from such a surface up to a solution of the relativistic Dirac equation in a domain. Formula for the square of the singular Cauchy‐type integral is given. The proofs of all these facts are based on intimate relations between time‐harmonic solutions of the relativistic Dirac equation and some versions of quaternionic analysis. Copyright © 2002 John Wiley & Sons, Ltd.