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Harmonic functions and harmonic forms are one of the most important objects of mathematical and theoretical physics. For example, they are widely used in field theory, determining the strength and induction of some electrostatic and magnetostatic fields. As a rule, harmonic functions are defined on multidimensional Euclidean and pseudo-Euclidean spaces. Due to the additional symmetries of the model, the specified objects must satisfy various conditions, such as homogeneity conditions.
This paper presents a method of constructing new homogeneous harmonic functions in the case of rational homogeneity index of the original function. No additional restrictions are imposed on the homogeneity index of both the original function and the new function constructed from it. As an example, we consider the case of constructing harmonic functions in two-dimensional Euclidean space.
Differential operators of the first and second orders with coefficients in the form of linear and quadratic forms are used to study the properties of such functions. The problem is reduced to the analysis of solvability conditions of linear partial differential equations of the first and second orders. Obtained method of constructing new homogeneous harmonic functions was based on the theory of analytic functions of a complex variable. The paper presents an example of construction of a harmonic function by the above method.
