Analog of the classical borel theorem for entire harmonic functions in ℝn and generalized orders - Veselovska, Olga (Eastern-European Journal of Enterprise Technologies, 2018)
The article describes research on the growth of functions that are harmonic in the whole space ℝn, 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 ℝn 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 ℝn. Besides, the study has revealed the most general characteristics of the growth of entire harmonic functions in ℝn 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.