# Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics

## Abstract

We formulate a boundary-value problem for eigenvalues and eigenfunctions of the Helmholtz equation in a complex domain with the use of mutually conjugated complex variables. The obtained systems of functions are orthogonal in this domain and constructed by using the Bessel functions and powers of the conformal mappings of the analyzed domains onto a circle. The solutions of boundary-value problems for the principal equations of mathematical physics (hyperbolic, parabolic and elliptic types) are obtained in the form of the sums of series in the systems of functions orthogonal over the domain.

- Authors:

- “L’vivs’ka Politekhnika” National University (Ukraine)

- Publication Date:

- OSTI Identifier:
- 22771555

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 229; Journal Issue: 2; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BESSEL FUNCTIONS; BOUNDARY-VALUE PROBLEMS; CONFORMAL MAPPING; EIGENFUNCTIONS; EIGENVALUES; MATHEMATICAL SOLUTIONS

### Citation Formats

```
Sukhorol’s’kyi, M. A..
```*Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3669-4.

```
Sukhorol’s’kyi, M. A..
```*Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics*. United States. doi:10.1007/S10958-018-3669-4.

```
Sukhorol’s’kyi, M. A.. Thu .
"Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics". United States. doi:10.1007/S10958-018-3669-4.
```

```
@article{osti_22771555,
```

title = {Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics},

author = {Sukhorol’s’kyi, M. A.},

abstractNote = {We formulate a boundary-value problem for eigenvalues and eigenfunctions of the Helmholtz equation in a complex domain with the use of mutually conjugated complex variables. The obtained systems of functions are orthogonal in this domain and constructed by using the Bessel functions and powers of the conformal mappings of the analyzed domains onto a circle. The solutions of boundary-value problems for the principal equations of mathematical physics (hyperbolic, parabolic and elliptic types) are obtained in the form of the sums of series in the systems of functions orthogonal over the domain.},

doi = {10.1007/S10958-018-3669-4},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 2,

volume = 229,

place = {United States},

year = {2018},

month = {2}

}

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