Abstract
The article describes research on the growth of functions that are harmonic in the whole space ℝ n , n≥3, and thus they are called entire harmonic. A relation has been established between the maximum terms of entire functions of finite order in the plane, which are given by power series whose coefficients are somewhat connected. Also, the maximum modulus of a harmonic function in the space ℝ n is evaluated through the maximum modulus of some entire function in the plane, the coefficients which are expressed in terms of the coefficients of the expansion of the harmonic function in a series by Laplace spherical functions. These results made it possible to obtain an analog of the classical Borel theorem for entire harmonic functions of finite order in ℝ n . Besides, the study has revealed the most general characteristics of the growth of entire harmonic functions in ℝ n in terms of the uniform norm of Laplace spherical functions in the expansion of harmonic functions in series. Slow growth of the harmonic functions in the space has also been studied. The obtained results are analogous to the classical results that are known for entire functions of one complex variable. The research findings are important because harmonic functions occupy a special place not only in many mathematical studies but also in the application of mathematical analysis to physics and mechanics, where these functions often describe various stationary processes.
Highlights
This is due to the fact that the potentials of the most important vector fields considered in physics are harmonic functions and any harmonic function can be considered as the potential of a certain field
There is no analog for harmonic functions in the case of the n-dimensional space, which are decomposed into series by Laplace spherical functions
We obtain a relation between the maximum terms of entire finite-order functions in the plane given by power series whose coefficients are somewhat connected
Summary
Harmonic functions, being a natural generalization of linear functions of one variable, are in some sense the simplest functions of several variables. It is important to express the generalized growth characteristics of harmonic functions in the space Rn in terms of the uniform norm of spherical Laplace functions that are included in the expansions of such functions in series as well as to establish an analog of the classical Borel theorem for harmonic functions of the n-dimensional space. Expressions for the order and type of solutions of some linear differential equations with partial derivatives in terms of the error of axisymmetric harmonic polynomial approximation and Lagrange interpolation are obtained in [19]. The characteristics of the growth of harmonic functions of the three-dimensional or n-dimensional spaces have been expressed in terms of the errors of approximation or interpolation of these functions by different polynomials in relation to different norms, as well as through the expansion coefficients of harmonic functions into series by the adjointed Legendre and Chebyshov polynomials or the norm of the gradient at the origin. There is no analog for harmonic functions in the case of the n-dimensional space, which are decomposed into series by Laplace spherical functions
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