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Journal articles on the topic 'Conformal mappings of special domains'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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1

Denega, Iryna. "Estimation of the products of the inner radii of domains with an additional symmetry condition." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 83–90. http://dx.doi.org/10.37069/1683-4720-2019-33-6.

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In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.

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2

Costamagna, Eugenio, and Paolo Di Barba. "Inhomogeneous dielectrics: conformal mapping and finite-element models." Open Physics 15, no. 1 (December 29, 2017): 839–44. http://dx.doi.org/10.1515/phys-2017-0099.

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AbstractField singularities in electrostatic and magnetostatic fields require special attention in field calculations, and today finite element methods are normally used, both in homogeneous and in inhomogeneous dielectric cases. Conformal mappings are a traditional tool in the homogeneous case, but two-stage Schwarz-Christoffel + Finite Difference procedures have been proposed for a long time to solve problems in case of inhomogeneous dielectric materials too. This allowed to overcome accuracy problems caused by convex corners in the domain boundary and relevant field singularities, and to easily apply finite difference (FD) solvers in rectangular domains. In this paper, compound procedures Schwarz-Christoffel + Finite Elements Method procedures are suggested, to improve both the accuracy and the speed of second stage calculations. The results are compared to Schwarz-Christoffel + Finite Difference and to direct finite-element calculations, and the small differences analyzed considering a well know case study geometry,i.e., a shielded dielectric-supported stripline geometry.

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3

Kalmoun, El Mostafa, Mohamed M. S. Nasser, and Khalifa A. Hazaa. "The Motion of a Point Vortex in Multiply-Connected Polygonal Domains." Symmetry 12, no. 7 (July 16, 2020): 1175. http://dx.doi.org/10.3390/sym12071175.

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We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

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4

Crowdy, D. G., and A. S. Fokas. "Conformal mappings to a doubly connected polycircular arc domain." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (May 31, 2007): 1885–907. http://dx.doi.org/10.1098/rspa.2007.1847.

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The explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by Akhiezer in 1928. The construction of an analogous formula for the case of a polycircular arc domain, i.e. for a doubly connected domain whose boundaries are a union of circular arc segments, has remained an important open problem. In this paper, we present this explicit formula. We first introduce a new method for deriving the classical formula of Akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain. As an analytical check of the formula, a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction. As an illustrative example, a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by Lord Rayleigh is considered in detail.

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5

CROWDY, DARREN. "CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS." ANZIAM Journal 53, no. 3 (January 2012): 171–89. http://dx.doi.org/10.1017/s1446181112000119.

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AbstractConformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

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6

Dittmar, Bodo. "The Robin function and conformal welding – A new proof of the existence." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 43–51. http://dx.doi.org/10.1515/gmj-2017-0057.

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AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

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7

Crowdy, Darren. "Stress fields around two pores in an elastic body: exact quadrature domain solutions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2180 (August 2015): 20150240. http://dx.doi.org/10.1098/rspa.2015.0240.

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Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.

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8

Maitani, Fumio. "Conformal slit mappings from periodic domains." Kodai Mathematical Journal 28, no. 2 (June 2005): 265–74. http://dx.doi.org/10.2996/kmj/1123767007.

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9

Herron, David A., and Pekka Koskela. "Quasiextremal distance domains and conformal mappings onto circle domains." Complex Variables, Theory and Application: An International Journal 15, no. 3 (September 1990): 167–79. http://dx.doi.org/10.1080/17476939008814448.

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10

Bourchtein, Andrei, and Ludmila Bourchtein. "On Conformal Conic Mappings of Spherical Domains." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/840280.

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The problem of the generation of homogeneous grids for spherical domains is considered in the class of conformal conic mappings. The equivalence between secant and tangent projections is shown and splitting the set of conformal conic mappings into equivalence classes is presented. The problem of minimization of the mapping factor variation is solved in the class of conformal conic mappings. Obtained results can be used in applied sciences, such as geophysical fluid dynamics and cartography, where the flattening of the Earth surface is required.

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11

刘, 子淇. "A Special Class of Conformal Mappings." Advances in Applied Mathematics 10, no. 09 (2021): 3010–24. http://dx.doi.org/10.12677/aam.2021.109315.

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12

Carroll, Tom, and J. B. Twomey. "Conformal Mappings of Close-to-Convex Domains." Journal of the London Mathematical Society 55, no. 3 (June 1997): 489–98. http://dx.doi.org/10.1112/s0024610797005188.

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13

Crowdy, Darren, and Jonathan Marshall. "Conformal Mappings between Canonical Multiply Connected Domains." Computational Methods and Function Theory 6, no. 1 (June 2006): 59–76. http://dx.doi.org/10.1007/bf03321118.

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14

Crowdy, Darren. "Calculating the lift on a finite stack of cylindrical aerofoils." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (January 24, 2006): 1387–407. http://dx.doi.org/10.1098/rspa.2005.1631.

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The classic exact solution due to Lagally (Lagally, M. 1929 Die reibungslose strömung im aussengebiet zweier kreise. Z. Angew. Math. Mech . 9 , 299–305.) for streaming flow past two cylindrical aerofoils (or obstacles) is generalized to the case of an arbitrary finite number of cylindrical aerofoils. Given the geometry of the aerofoils, the speed and direction of the oncoming uniform flow and the individual round-aerofoil circulations, the complex potential associated with the flow is found in analytical form in a parametric pre-image region that can be conformally mapped to the fluid region. A complete determination of the flow then follows from knowledge of the conformal mapping between the two regions. In the special case where the aerofoils are all circular, the conformal mapping from the parametric pre-image region to the fluid domain is a Möbius mapping. The solution for the complex potential in such a case can then be used, in combination with the Blasius theorem, to compute the distribution of hydrodynamic forces on the multi-aerofoil configuration.

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15

Berezovski, Volodymyr, Yevhen Cherevko, and Lenka Rýparová. "Conformal and Geodesic Mappings onto Some Special Spaces." Mathematics 7, no. 8 (July 25, 2019): 664. http://dx.doi.org/10.3390/math7080664.

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In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces. The main equations for the mappings are obtained as a closed system of Cauchy-type differential equations in covariant derivatives. We find the number of essential parameters which the solution of the system depends on. A similar approach was applied for the case of conformal mappings of Riemannian spaces onto Ricci-m-symmetric Riemannian spaces, as well as geodesic mappings of spaces with affine connections onto Ricci-m-symmetric spaces.

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16

Koşar, Cem. "Simultaneous approximation of conformal mappings on smooth domains." Filomat 33, no. 1 (2019): 281–87. http://dx.doi.org/10.2298/fil1901281k.

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17

Oktay, Burcin, and Daniyal M. Israfilov. "An approximation of conformal mappings on smooth domains." Complex Variables and Elliptic Equations 58, no. 6 (June 2013): 741–50. http://dx.doi.org/10.1080/17476933.2011.617009.

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18

Lesley, Frank David. "Conformal mappings of domains satisfying a wedge condition." Proceedings of the American Mathematical Society 93, no. 3 (March 1, 1985): 483. http://dx.doi.org/10.1090/s0002-9939-1985-0774007-0.

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19

Bourchtein, Ludmila, and Andrei Bourchtein. "One inequality for conformal mappings of spherical domains." Mathematical Inequalities & Applications, no. 2 (2006): 233–46. http://dx.doi.org/10.7153/mia-09-24.

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20

Bourchtein, Andrei, and Ludmila Bourchtein. "Some problems of conformal mappings of spherical domains." Zeitschrift für angewandte Mathematik und Physik 58, no. 6 (November 29, 2006): 926–39. http://dx.doi.org/10.1007/s00033-006-5126-3.

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21

Gutlyanskii, V. Ya, and S. A. Kopanev. "Conformal mappings of the disk onto convex domains." Ukrainian Mathematical Journal 44, no. 10 (October 1992): 1217–23. http://dx.doi.org/10.1007/bf01057677.

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22

Crowdy, Darren G., Athanassios S. Fokas, and Christopher C. Green. "Conformal Mappings to Multiply Connected Polycircular Arc Domains." Computational Methods and Function Theory 11, no. 2 (January 2012): 685–706. http://dx.doi.org/10.1007/bf03321882.

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23

Samsonia, Z., and L. Zivzivadze. "Conformal and Quasiconformal Mappings of Close Multiply-Connected Domains." gmj 9, no. 2 (June 2002): 367–82. http://dx.doi.org/10.1515/gmj.2002.367.

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Abstract Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

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24

RADYGIN, Vladimir Mikhailovich, and Ivan Sergeyevich POLANSKY. "MODIFIED METHOD OF SUCCESSIVE CONFORMAL MAPPINGS OF POLYGONAL DOMAINS." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 39(1) (March 1, 2016): 25–35. http://dx.doi.org/10.17223/19988621/39/3.

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25

Avkhadiev, Farit Gabidinovich, and Pavel Leonidovich Shabalin. "Conformal mappings of circular domains on finitely-connected non-Smirnov type domains." Ufimskii Matematicheskii Zhurnal 9, no. 1 (2017): 3–17. http://dx.doi.org/10.13108/2017-9-1-3.

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26

CROWDY, DARREN, and AMIT SURANA. "Contour dynamics in complex domains." Journal of Fluid Mechanics 593 (November 23, 2007): 235–54. http://dx.doi.org/10.1017/s002211200700866x.

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This paper demonstrates that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected planar domain bounded by impenetrable walls. A general numerical scheme is presented based on this formulation. The algorithm makes use of conformal mappings and follows the evolution of a conformal pre-image of a given vortex patch in a canonical multiply connected circular pre-image region. The evolution of vortex patches can be computed given just the conformal map from this pre-image region to the physical fluid region. The efficacy of the scheme is demonstrated by illustrative examples.

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27

N. Ivanshin, Pyotr, and . "Continued Fractions and Conformal Mappings for Domains with Angel Points." International Journal of Engineering & Technology 7, no. 4.7 (September 27, 2018): 409. http://dx.doi.org/10.14419/ijet.v7i4.7.23039.

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Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.

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28

Kovalev, Leonid V., and Liulan Li. "On the existence of harmonic mappings between doubly connected domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 3 (April 22, 2018): 619–28. http://dx.doi.org/10.1017/s0308210517000506.

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While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.

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29

Dym, Nadav, Raz Slutsky, and Yaron Lipman. "Linear variational principle for Riemann mappings and discrete conformality." Proceedings of the National Academy of Sciences 116, no. 3 (December 28, 2018): 732–37. http://dx.doi.org/10.1073/pnas.1809731116.

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We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: It is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in H1, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the discrete Riemann mappings between each Lipschitz domain and the triangle.

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30

Ivanshin, Pyotr N. "Continued Fractions and Conformal Mappings for Domains with Angle Points." International Journal of Psychosocial Rehabilitation 23, no. 3 (September 20, 2019): 712–33. http://dx.doi.org/10.37200/ijpr/v23i3/pr190361.

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31

Garnett, John B., and Shan Shuang Yang. "Quasiextremal distance domains and integrability of derivatives of conformal mappings." Michigan Mathematical Journal 41, no. 2 (1994): 389–406. http://dx.doi.org/10.1307/mmj/1029005004.

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32

Tsuchiya, Takuya. "Finite Element Approximation of Conformal Mappings to Unbounded Jordan Domains." Numerical Functional Analysis and Optimization 35, no. 10 (July 21, 2014): 1382–97. http://dx.doi.org/10.1080/01630563.2013.837482.

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33

Vuorinen, Matti. "On quasiregular mappings and domains with a complete conformal metric." Mathematische Zeitschrift 194, no. 4 (December 1987): 459–70. http://dx.doi.org/10.1007/bf01161915.

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34

Denisov, V. I. "Special conformal mappings in the general theory of relativity." Journal of Soviet Mathematics 48, no. 1 (January 1990): 36–40. http://dx.doi.org/10.1007/bf01098040.

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35

Ben Boubaker, Mohamed Amine, and Mohamed Selmi. "Sharp Estimates for Green’s Functions of Cone-Type Planar Domains." Journal of Complex Analysis 2016 (February 10, 2016): 1–11. http://dx.doi.org/10.1155/2016/7208285.

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We establish sharp estimates for Green’s functions of cone-type planar domains. Our work generalizes all estimates given by Zhao in 1988 and Selmi in 2000. Our principal idea is to use conformal mappings.

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36

Inoue, Tetsuo, Hideo Kuhara, Kaname Amano, and Dai Okano. "Experiment on numerical conformal mapping of unbounded multiply connected domain in fundamental solutions method." International Journal of Mathematics and Mathematical Sciences 26, no. 1 (2001): 55–63. http://dx.doi.org/10.1155/s0161171201004112.

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We are concerned with the experiment on numerical conformal mappings. A potentially theoretical scheme in the fundamental solutions method, different from the conventional one, has been recently proposed for numerical conformal mappings of unbounded multiply connected domains. The scheme is based on the asymptotic theorem on extremal weighted polynomials. The scheme has the characteristic called invariant and dual. Applying the scheme for typical examples, we will show that the numerical results of high accuracy may be obtained.

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37

Dolzhenko, E. P., and S. V. Kolesnikov. "Boundary behavior of derivatives of conformal mappings of simply connected domains." Moscow University Mathematics Bulletin 69, no. 5 (September 2014): 205–10. http://dx.doi.org/10.3103/s0027132214050052.

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38

Pearce, Kent. "A Constructive Method for Numerically Computing Conformal Mappings for Gearlike Domains." SIAM Journal on Scientific and Statistical Computing 12, no. 2 (March 1991): 231–46. http://dx.doi.org/10.1137/0912013.

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39

Vlasov, V. I., and A. B. Pal’tsev. "An analytical-numerical method for conformal mappings of complex-shaped domains." Doklady Mathematics 80, no. 3 (December 2009): 790–92. http://dx.doi.org/10.1134/s1064562409060027.

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40

Yang, Shanshuang. "Conformal invariants of smooth domains and extremal quasiconformal mappings of ellipses." Illinois Journal of Mathematics 41, no. 3 (September 1997): 438–52. http://dx.doi.org/10.1215/ijm/1255985738.

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41

Hakula, Harri, Tri Quach, and Antti Rasila. "The Conjugate Function Method and Conformal Mappings in Multiply Connected Domains." SIAM Journal on Scientific Computing 41, no. 3 (January 2019): A1753—A1776. http://dx.doi.org/10.1137/17m1124164.

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42

Najdanovic, Marija, Milan Zlatanovic, and Irena Hinterleitner. "Conformal and geodesic mappings of generalized equidistant spaces." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 71–84. http://dx.doi.org/10.2298/pim1512071n.

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We consider conformal and geodesic mappings of generalized equidistant spaces. We prove the existence of mentioned nontrivial mappings and construct examples of conformal and geodesic mapping of a 3-dimensional generalized equidistant space. Also, we find some invariant objects (three tensors and four which are not tensors) of special geodesic mapping of generalized equidistant space.

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43

Kırık, Bahar, and Özen Zengin. "Conformal mappings of quasi-Einstein manifolds admitting special vector fields." Filomat 29, no. 3 (2015): 525–34. http://dx.doi.org/10.2298/fil1503525k.

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As it is known, Einstein manifolds play an important role in geometry as well as in general relativity. Einstein manifolds form a natural subclass of the class of quasi-Einstein manifolds. In this work, we investigate conformal mappings of quasi-Einstein manifolds. Considering this mapping, we examine some properties of these manifolds. After that, we also study some special vector fields under this mapping of these manifolds and some theorems about them are proved.

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44

Bourchtein, Ludmila. "Conformal mappings of multiply connected domains onto canonical domains using the Green and Neumann functions." Complex Variables and Elliptic Equations 58, no. 6 (June 2013): 821–36. http://dx.doi.org/10.1080/17476933.2011.622045.

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45

Jones, Gareth Wyn, and L. Mahadevan. "Planar morphometry, shear and optimal quasi-conformal mappings." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2153 (May 8, 2013): 20120653. http://dx.doi.org/10.1098/rspa.2012.0653.

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To characterize the diversity of planar shapes in such instances as insect wings and plant leaves, we present a method for the generation of a smooth morphometric mapping between two planar domains which matches a number of homologous points. Our approach tries to balance the competing requirements of a descriptive theory which may not reflect mechanism and a multi-parameter predictive theory that may not be well constrained by experimental data. Specifically, we focus on aspects of shape as characterized by local rotation and shear, quantified using quasi-conformal maps that are defined precisely in terms of these fields. To make our choice optimal, we impose the condition that the maps vary as slowly as possible across the domain, minimizing their integrated squared-gradient. We implement this algorithm numerically using a variational principle that optimizes the coefficients of the quasi-conformal map between the two regions and show results for the recreation of a sample historical grid deformation mapping of D’Arcy Thompson. We also deploy our method to compare a variety of Drosophila wing shapes and show that our approach allows us to recover aspects of phylogeny as marked by morphology.

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46

Gauthier, Paul M., and Fatemeh Sharifi. "The Carathéodory Reflection Principle and Osgood–Carathéodory Theorem on Riemann Surfaces." Canadian Mathematical Bulletin 59, no. 4 (December 1, 2016): 776–93. http://dx.doi.org/10.4153/cmb-2016-051-1.

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AbstractThe Osgood–Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

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47

Włodarczyk, Kazimierz. "Area methods, extremal problems and extremal domains for pairs of conformal mappings." Annales Polonici Mathematici 45, no. 3 (1985): 203–11. http://dx.doi.org/10.4064/ap-45-3-203-211.

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48

Dolzhenko, E. P. "On the boundary smoothness of conformal mappings between domains with nonsmooth boundaries." Doklady Mathematics 76, no. 1 (August 2007): 514–18. http://dx.doi.org/10.1134/s1064562407040096.

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49

Nasser, Mohamed M. S. "PlgCirMap: A MATLAB toolbox for computing conformal mappings from polygonal multiply connected domains onto circular domains." SoftwareX 11 (January 2020): 100464. http://dx.doi.org/10.1016/j.softx.2020.100464.

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50

Kosiński, Łukasz. "Proper holomorphic mappings in the special class of Reinhardt domains." Annales Polonici Mathematici 92, no. 3 (2007): 285–97. http://dx.doi.org/10.4064/ap92-3-7.

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