Complete Systems of Beltrami Fields Using Complex Quaternions and Transmutation Theory

Abstract

Beltrami fields are complex vector fields $${\mathbf {F}}$$ which satisfy the equation $${\text {curl}} {\mathbf {F}} + \lambda {\mathbf {F}}=0.$$ Such fields appear in astrophysics, electromagnetics and plasma physics. We construct a complete system of solutions to the differential equation $$(D+\lambda (x_3)+M^{\gamma (x_3){\mathbf {e}}_3})u=0$$ for a complex quaternionic valued function u in a symmetric domain in $${\mathbb {R}}^3$$ , by means of transmutation operators. We then apply this result to construct Beltrami fields, giving a complete system of fields when $$\lambda $$ depends only on $$x_3$$ .