Representation of Weierstrass integral via Poisson integrals

Abstract

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.