General approach to constructing solutions of boundary value problems for Helmholtz's equations is considered. By transforming coordinates applying conforming mappings of corresponding domains onto the circle, a set of solutions of Helmholtz's equation in different coordinate systems is obtained. Solutions of boundary value problems for this equation in the plane with an elliptical hole are constructed.

- Lavrentyev M. A., Shabat B. V. Methods of the theory of functions of a complex variable. Moscow: Nauka (1987), (in Russian).
- Korn G. A., Korn T. M. Mathematical Handbook for Scientists and Engineers. Dover publications, Inc: Mineola, New York (2000).
- Sidorov V., Fedoryuk M., Shabunin M. Lectures on the theory of functions of a complex variable. Moscow: Nauka (1982), (in Russian).
- Sukhorolsky M. A. Systems of Helmholtz equation solutions. Bulletin of Lviv Polytechnic National University. Series Physical and mathematical sciences.
**718**, 19–34 (2011), (in Ukrainian). - Sukhorolsky M. A. Analytical solutions of the Helmholtz equation. Mathematical problems of mechanics of inhomogeneous structures. Ed. by Lukovskiy I., Kit G., Kushnir R. Lviv: IAPMM of NAS of Ukraine. 160–163 (2014), (in Ukrainian).
- Sukhorolsky M. A., Kostenko I. S., Dostoyna V. Construction of solutions of partial diﬀerential equations in the form of contour integrals. Bulletin of KNTU.
**2**(47), 323–326 (2013), (in Ukrainian). - Markushevich A. I. Theory of analytic functions. V.2. Moscow: Nauka (1968), (in Russian).
- Paszkowski S. Zastosowania numeryczne wielomianow i szeregow Czebyszewa. Warszawa: Panstwowe Wydawnictwo Naukowe (1975).

Math. Model. Comput. Vol. 1, No. 2, pp. 256-263 (2014)