Conclusions

In this paper, we present a different way of obtaining solutions to the Helmholtz equation without imposing boundary conditions, that is, the solutions are not necessarily limited to certain domains of the complex plane $\mathbb{C}$. These solutions depend on three holomorphic functions. Only when $c$ is negative because of the regularity of the solution, we need to restrict the solution to a certain domain of $\mathbb{C}$ (see Figures 4.5, 4.6, 5.7 and 5.8).