Boundary values and zeros of Harmonic Product of Complex-valued Harmonic Functions in a simply connected bounded Domain (2503.08105v1)

Abstract: The product of two complex-valued harmonic function is not in general complex-valued harmonic function. In this paper we show that if a complex-valued harmonic function is the product of two complex-valued harmonic functions, then it is the difference of two squares, one is analytic and the other is co-analytic. As a result of this, if one of the factors of the product is known, then the other factor is expressed in terms of the known factor explicitly. As an application of this we determine the boundary value of one of the factors and the product provided the boundary value of the other factor is known. It is shown that the boundary value of the product is a pure imaginary constant. Moreover, if such a product and factors are complex-valued harmonic polynomials, then the number of zeros of the product is at most half of the square of its degree which is by half smaller than what is known in the literature about the maximum number of zeros of complex-valued harmonic polynomials. The feature of this paper is to explore multipliers for some subspace of complex-valued harmonic functions and determine some nontrivial invariant subspace.