Some properties of a~regular solution of the Helmholtz equation in a~two-dimensional domain

V. A. Il'in

Matematičeskie zametki, Tome 15 (1974) no. 6, pp. 885-890.

Résumé

In this paper we prove that if the function $u_\lambda$ is a regular solution of the equation $\Delta_2u+\lambda u=0$ in an arbitrary two-dimensional domain $g$ and if at an arbitrary point $M$ of the domain $g$ we introduce polar coordinates $r$ and $\varphi$, then for an arbitrary value of the polar radius $r$, less than the distance of the point $M$ from the boundary of the domain $g$, the following formula is valid: $$ \int_0^{2\pi}u_\lambda(r,\varphi)e^{in\varphi}\,d\varphi=2\pi(\sqrt\lambda)^{-n}J_n(r\sqrt\lambda)\Bigl(\frac\partial{\partial x}+i\frac\partial{\partial y}\Bigr)^nu_\lambda(M). $$ Simultaneously, we show that the derivative $\frac{\partial^nu_\lambda(0,\varphi)}{\partial r^n}$ is an $n$-th order trigonometric polynomial.