### Systems of Powers of Conformal Mappings and Conjugate Systems of Functions

#### Abstract

Examples of the biorthogonal systems whose elements are the compositions of fractional rational and exponential functions are considered. We also construct solutions to the boundary value problems for the Helmholtz equation in the case when the boundary functions is defined by series in terms of biorthogonal systems of functions.

#### Keywords

#### References

Dzyadyk V.K. textit{Introduction to the theory of uniform approximation of functions by polynomials.} -- Moscow: Nauka, 1977, -- 512 p. (in Russian).

Leontyev A. F. textit{Sequences of polynomials from exponents.} -- Moscow: Nauka, 1980, -- 384 p. (in Russian).

Markushevich A. I. textit{Theory of analytic functions.}-- Moscow: Nauka, 1968, V.2, -- 624 p. (in Russian).

Smirnov V. I., Lebedyev N. A. textit{Constructional theory of the functions of a complex variable.} -- Moscow: Nauka, 1964, -- 440 p. (in Russian).

Sukhorolsky M. A. textit{Expansion of analytical functions by the system of polynomials of Mellin type.} Bulletin of Lviv Polytechnic National University. Series Physical and Mathematical Sciences. -- 2005. -- No. 346. -- P. 111 -- 115. (in Ukrainian).

Sukhorolsky M. A. textit{Development of functions by the system of polynomials biortho-gonal on the closed contour with the system of regular functions in infinitely remote point.} Ukrainian Mathematical Journal. -- 2010. -- V.62, No. 2. -- P. 238 -- 254. (in Ukrainian).

Korn G. A., Korn T. M. textit{Mathematical Handbook for Scientists and Engineers.} -- New York: Dover Publications, Inc: Mineola, 2000, -- 1130 p. (in USA)

Sukhorolsky M. A. textit{Analytical solutions to Helmholtz's equation.} Mathematical problems of mechanics of inhomogeneous structures. Ed. by Lukovskiy I., Kit J., Kushnir R., Lviv: IAPMM of NAS of Ukraine. -- 2014. -- P. 160 -- 163. (in Ukrainian).

Sukhorolsky M. A., Dostoyna V. V. textit{One class of biorthogonal systems of functions that arise in the solution of the Helmholtz's equation in the cylindrical coordinate system.} J. Math. Sci. -- 2013. -- V. 192, No. 5. -- P. 541 -- 554. (in Ukrainian).

Lavrentyev M. A., Shabat B. V. textit{Methods of the theory of functions of a complex variable.} -- Moscow: Nauka, 1987, -- 698 p. (in Russian).

### Refbacks

- There are currently no refbacks.