TY - JOUR
T1 - Towards a quaternionic function theory linked with the Lamé's wave functions
AU - Morais, J.
AU - Pérez-De La Rosa, M. A.
N1 - Publisher Copyright:
Copyright © 2015 John Wiley & Sons, Ltd.
PY - 2015/11/30
Y1 - 2015/11/30
N2 - Over the past few years, considerable attention has been given to the role played by the Lamé's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lamé's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non-commutative framework. We show that the theory of the LQWFs is determined by the Moisil-Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lamé's wave equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel-Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski-Plemelj formulae, the Dα-hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs.
AB - Over the past few years, considerable attention has been given to the role played by the Lamé's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lamé's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non-commutative framework. We show that the theory of the LQWFs is determined by the Moisil-Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lamé's wave equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel-Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski-Plemelj formulae, the Dα-hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs.
KW - Cauchy-type integral
KW - Helmholtz equation
KW - Lamé's wave functions
KW - Quaternionic analysis
KW - Sokhotski-Plemelj formulae
KW - prolate and oblate spheroidal wave functions
KW - spherical wave functions
UR - http://www.scopus.com/inward/record.url?scp=84959327949&partnerID=8YFLogxK
U2 - 10.1002/mma.3376
DO - 10.1002/mma.3376
M3 - Artículo
SN - 0170-4214
VL - 38
SP - 4365
EP - 4387
JO - Mathematical Methods in the Applied Sciences
JF - Mathematical Methods in the Applied Sciences
IS - 17
ER -