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REVIEW 2 major objections 4 minor 45 references

Computation of conformal invariants

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Pith's one-line read One integral solver computes conformal capacity to 14 digits.

desk verdict A solid, well-validated computational paper; the elliptic-capacity conjugate worry does not survive reading, but the non-symmetric benchmark could be tightened. read the letter →

arxiv 1908.04533 v2 pith:O42JQVTI submitted 2019-08-13 math.CV

Mohamed M S Nasser , Matti Vuorinen This is my paper · ORCID
classification math.CV MSC 65E0530C8531A15
keywords conformalcapacitygeneralizedNeumannkernelboundaryintegralequationsdoublyconnecteddomainshyperbolicellipticnumericalmappingcondensers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to make conformal invariants numerically routine. Its central claim is that one boundary integral equation, based on the generalized Neumann kernel, can map a ring-shaped domain conformally onto an annulus $\{q<|z|<1\}$ and thereby compute its capacity as $2\pi/\log(1/q)$. The same pipeline, with small auxiliary maps attached, handles smooth curves, polygons, slits, segment-circle rings, and half-plane slit domains. In tests with known exact values, the reported relative errors are near $10^{-14}$ for smooth boundaries and several correct digits for cornered boundaries. The same computed $q$ also gives hyperbolic capacity ($q$) and elliptic capacity ($\sqrt{q}$) for compact connected sets in the unit disk.

What carries the argument

The central object is the generalized Neumann kernel $N(s,t)$, a continuous kernel built from the boundary parametrization and an auxiliary function $A$; with its singular companion $M(s,t)$ it forms the integral equation $(I-N)\rho=-M\gamma$, which is uniquely solvable. The load-bearing identity is $q=e^{h_2-h_1}$, where $h_1,h_2$ are the two values of the piecewise constant function $h=[M\rho-(I-N)\gamma]/2$ on the two boundary components. This identity turns the numerical solution of a linear integral equation into the modulus of the annulus conformally equivalent to the given domain, and therefore into the capacity $2\pi/\log(1/q)$. The paper packages that whole computation in one function, `annq`, whose output feeds every example in the text.

What would settle it

Use the confocal-ellipse ring with $r_1>r_2>1$, for which the exact capacity is $2\pi/\log(r_1/r_2)$; with $r_2=2$, $r_1=4$, and $n=2^{12}$ discretization points per boundary, the method should give a relative error near $10^{-14}$. If it does not, the claimed smooth-boundary accuracy fails.

Watch

Extended reading notes

Core claim

The paper establishes that the conformal capacity, hyperbolic capacity, and elliptic capacity of planar domains can all be extracted from a single computation: the conformal modulus $q$ of an associated doubly connected domain. Solving the generalized Neumann kernel integral equation on the two boundary curves yields a piecewise constant function $h$, and the identity $q=e^{h_2-h_1}$ gives the modulus of the annulus to which the domain maps; conformal invariance then forces $\mathrm{cap}(G)=2\pi/\log(1/q)$. For a compact connected set $E\subset\mathbb{D}$, the same $q$ with $G=\mathbb{D}\setminus E$ is the hyperbolic capacity, and with $G$ the region between $E$ and its antipodal set $E^*$, $\sqrt{q}$ is the elliptic capacity. Slit and half-plane boundaries are reduced to smooth rings by elementary maps or an iterative pre-mapping, and the same solver applies. Exact-formula comparisons show relative errors near $10^{-14}$ for smooth boundaries, and for cornered boundaries the values match known benchmarks to several digits.

Load-bearing premise

For slit and half-plane geometries, the capacity result relies on an auxiliary numerical pre-mapping that converts the slit into a smooth curve, and the paper does not analyze that pre-mapping's error.

Editorial extensions

If this is right

  • A new doubly connected geometry needs only a boundary parametrization; the same solver returns its capacity, hyperbolic capacity, or elliptic capacity without a new derivation.
  • For cornered geometries such as square-in-square, polygon-in-polygon, and rectangle-slits, the method produces stable values that in the tested tables match exact or benchmark values to several digits, so it can be used as a cross-check for other numerical schemes.
  • The $O(n \log n)$ cost and reported runtimes below roughly ten seconds make parameter studies feasible, such as tracing capacity as a slit moves or as a boundary component changes shape.
  • The inequality $\mathrm{cap_e}(E)\le \mathrm{cap_h}(E)$ for compact connected subsets of the unit disk becomes numerically checkable for arbitrary shapes, as demonstrated for the non-symmetric amoeba region.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since several other conformal invariants, such as extremal length and moduli of curve families in doubly connected domains, are also functions of the annulus modulus $q$, the same computed $q$ could supply those values with no additional integral solving.
  • A graded-mesh version of the trapezoidal rule, which the paper notes is needed at corners, should close the accuracy gap seen for square and polygon boundaries while keeping the same pipeline.
  • The unquantified pre-mapping error for slit domains could be measured empirically by comparing the iterative pre-mapping results with the exact formulas for two-real-slits or half-plane slits.
  • Because geometry enters only through the boundary parametrization, coupling the solver with automatic differentiation could produce geometric gradients of capacity, enabling shape optimization.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper presents a boundary integral equation method, based on the generalized Neumann kernel, for computing the conformal capacity of doubly connected planar domains by numerically constructing the conformal map onto a circular annulus. The authors supply a MATLAB implementation, annq, and validate it against exact formulas for several model geometries: two confocal ellipses, a square-in-square domain, two collinear slits, a segment and a circle, and a half-plane with a vertical slit. They report relative errors around 1e-14 for smooth boundaries and roughly twelve correct digits for corner domains. The method is then applied to slit and polygon geometries and to the numerical computation of hyperbolic and elliptic capacities of compact sets in the unit disk.

Significance. If the advertised claims are fully supported, the paper provides a useful, versatile, and easily modifiable computational tool for conformal invariants. Its strengths are the independent exact-formula benchmarks, the explicit MATLAB code, and the fact that the reported errors are not fitted to the benchmarks. The ring-capacity and hyperbolic-capacity results are convincing. However, the elliptic-capacity branch, which is part of the advertised scope, is currently undermined by an incorrect definition of the antipodal set in Section 8.2 and by a consequently invalid numerical test in Section 8.5. The central computational method itself appears sound, and the defect is localized and fixable.

major comments (2)
  1. [Section 8.2, Eqs. (47)-(48), and Section 8.5, Table 12] The definition of the antipodal set E* = {-1/a : a in E} is not the involution used in the Duren-Kuhnau elliptic capacity theory cited as [13]. The elliptic capacity of a set in the disk is associated with the fixed-point-free anti-Mobius antipodal map a -> -1/\bar a; the map a -> -1/a printed in the paper is a different holomorphic involution with fixed points ±i, and the two definitions agree only for sets invariant under complex conjugation. As a consequence, for the non-conjugation-symmetric amoeba of Section 8.5 the doubly connected domain between E and the incorrectly defined E* is not the annulus whose modulus yields cape(E), and the values labeled cape(E) in Table 12 together with the reported numerical confirmation of cape(E) < caph(E) do not test the stated inequality. The definition also breaks down when 0 belongs to E. Please replace E* by {-1/\bar a : a in E}, interpreted on the Riemann sphere when needed, recompute Section 8.5, and ensure that the code underlying annq uses the corrected definition.
  2. [Sections 4 and 6] The general two-slit and half-plane-with-slit computations rely on the iterative method of [33] to construct a smooth doubly connected model domain, but the paper does not analyze the convergence or the accuracy of this auxiliary step. The benchmarks in Sections 4.1 and 6 with exact formulas show that the composed method is accurate in those cases, including the iterative step; however, for the genuinely non-symmetric configurations in Section 4.3 and Figure 16 no error estimate or refinement study for the auxiliary mapping is supplied. Because the advertised scope includes slit geometries, the authors should either quantify the error introduced by this step, for example by varying the stopping criterion of the iteration or comparing successive iterates, or explicitly state that these specific results are benchmark-dependent.
minor comments (4)
  1. [Sections 2.3 and 7.3] There are several typographical errors: "using using MATLAB R2017a" in Section 2.3, "In view of of (7)" near Eq. (8), and "the the auxiliary map" in Section 7.3.
  2. [Section 8.3, Figure 21] The caption of Figure 21 mentions a quantity c(E) that is not defined in the text; it should be defined or removed.
  3. [Section 7.1, Table 9] The "exact" capacity at the bottom of Table 9 is itself obtained by numerically solving equations (43) with Mathematica; the reported agreement of 1.5e-9 is several orders of magnitude larger than the other errors in the paper, so the precision of the Mathematica solution and the source of this discrepancy should be stated.
  4. [Table 5] The header "z2\z1" is confusing because the row entries appear to be values of z2 while the columns are values of z1; the layout should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: the BIE solver is validated against independent exact formulas, and no fitted quantity is renamed as a prediction.

full rationale

I find no circular step that reduces an output to an input by construction. The paper's capacity computation reduces to computing the modulus q of a doubly connected domain by solving the boundary integral equation (9) and applying q = exp(h2 - h1) and cap(G) = 2*pi/log(1/q). Theorems 1 and 2, quoted from the authors' earlier work [28], state that q is obtained from the solution of the integral equation; this is a mathematical reduction, not a fit: the functions h and rho are obtained by solving an operator equation, and the exact benchmark values (confocal ellipses via the Joukowski map, square-in-square via elliptic integrals, two real segments via [44], segment-circle via [44]) are independent formula evaluations, not values used to determine any parameter. No parameter is fitted to a subset of capacity data and then "predicted" on a closely related quantity. The self-citations [28]-[30] and [33] supply the numerical method, the fbie implementation, and the iterative auxiliary mapping; they are load-bearing in the sense that the implementation is reused, but they are published, externally testable methods and do not encode the benchmark capacities or the claimed numerical results. The hyperbolic- and elliptic-capacity sections invoke the external theorems of Duren and Kuhnau [13] relating caph(E) = q and cape(E) = sqrt(q) to moduli of auxiliary domains; these are used as theoretical inputs, not as circular definitions, since the transfinite-diameter definitions (45) and (47) are stated separately. The reported inequality cape(E) < caph(E) for the amoeba-shaped set is obtained from two independent BIE computations and is not imposed by any fitted parameter. The skeptical note about the antipodal set E* = {-1/a} omitting the complex conjugate, if correct, is a mathematical correctness concern about the elliptic-capacity example, not a circularity: it does not show that a claimed output is equivalent by construction to an input. Accordingly, no circular step with the required quote-and-reduction evidence is present, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The method depends on established theorems on conformal annulus mapping, the generalized Neumann kernel, conformal invariance of capacity, and the iterative preimage construction from [33]. The exact benchmark formulas are external validation inputs, not assumptions of the method.

assumptions (5)
  • standard math Every doubly connected domain (ring) can be conformally mapped onto an annulus {q < |z| < 1}.
    Classical result used throughout; invoked in Section 2.6 for capacity via conformal invariance.
  • standard math The generalized Neumann kernel integral equation (I-N)rho = -M gamma is uniquely solvable and yields a piecewise constant h.
    Proved in [29]; the paper relies on it for the numerical method in Section 2.2.
  • standard math Capacity is invariant under conformal mappings.
    Used to reduce capacity of any ring to that of an annulus; stated in Section 2.1.
  • domain assumption For slit and half-plane domains, the iterative method [33] produces a conformally equivalent domain bordered by smooth Jordan curves.
    The paper does not prove or analyze this preimage step; it underlies Sections 4, 5, and 6.
  • standard math Hyperbolic capacity caph(E) equals q and elliptic capacity cape(E) equals sqrt(q) for a domain whose conformal modulus is q.
    Quoted from [13]; used in Section 8.

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Cite this review

Pith. "Pith review of Computation of conformal invariants." pith.science (2026). https://pith.science/paper/O42JQVTI

@misc{pith2026190804533,
  author       = {Pith},
  title        = {Pith review of: Computation of conformal invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O42JQVTI}},
  note         = {Machine review of arXiv:1908.04533}
}
read the original abstract

We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains are assumed to have smooth or piecewise smooth boundaries. The method we use is based on the boundary integral equation method developed and implemented in [30]. A characteristic feature of this method is that, with small changes in the code, a wide spectrum of problems can be treated and we include code snippets within the text to indicate implementation details. We compare the performance and accuracy to previous results in the cases when numerical data is available and also in the case of several model problems where exact results are available.

Figures

Figures reproduced from arXiv: 1908.04533 by the authors.

Figure 1
Figure 1. An example of a bounded doubly connected domain (left) an [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Results for the two confocal ellipses domain: Relative erro [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The polygon in polygon domain G for m = 5 and q = 0.5. The figure shows also the annulus q < |z| < 1 where the vertices of the two polygons are on the circles |z| = 1 and |z| = q. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: The two segments domain Ω for a = 0, b = 1, c = 1 − ı and d = 3 + 2ı (left) and the preimage domain G bordered by ellipses (right). 4.1 Two segments on the real axis When E = [0, 1] and F = [c, d] with d > c > 1, the exact capacity of Ω is known and is given by [44, 5.…
Figure 5
Figure 5. Figure 5: Relative errors of the computed conformal capacity for [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Results for the two segments domain: the contour lines of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Results for the two segments domain: the contour lines of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: The segment and circle domain Ω for a = 2 and r = 0.9 (left); and the image of this unbounded domain under the mapping ζ = Ψ−1 (z) (right). To apply our method presented in Section 2, we shall use first elementary mappings to map the domain Ω onto a domain G of the typ…
Figure 9
Figure 9. Figure 9: Results for the segment and circle domain: Relative errors [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: The domain Gr with c = 1.5, d = 3.5 for r = 0 (left), r = 0.1 (center) and r = b (right). It is clear from the definition of the closed set Fr that F0 ⊆ Fr ⊆ Fb for 0 ≤ r ≤ b. As r changes continuously from 0 to b, the closed set Fr changes continuously from the segme…
Figure 11
Figure 11. Figure 11: The values of cap(E, Fr) for c = 1.5, d = 3.5, 0 ≤ r ≤ b. 5.3 Segment and polygon In this example, we consider the doubly connected domain Ω in the exterior of the segment [0, 1] and a polygon with m vertices where m ≥ 3. We assume that the vertices of the polygon are…
Figure 12
Figure 12. Figure 12: The segment and polygon domain Ω for a = 1.6 and r = 0.5 (left); and the image of this unbounded domain under the mapping ζ = Ψ−1 (z) (right) [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Results for the segment and polygon domain: The comput [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: The half-plane with a segment domain Ω for [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Results for the half-plane with a segment domain: Relative [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Results for the half-plane with a segment domain: the con [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: The domain G in the exterior of the rectangular closed sets for d = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: The domain G in the exterior of a vertical rectangular closed set in the upper half-plane (left) and its image Gˆ under the auxiliary map Ψ (right) for d = 0.1. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: The domain G in the exterior of a horizontal rectangular closed set in the upper half-plane (left) and its image Gˆ under the auxiliary map Ψ (right) for d = 0.1. As in the previous example, the the auxiliary map w = Ψ(z) in (44) is used to transform the domain G onto…
Figure 20
Figure 20. Figure 20: The domain G for computing the hyperbolic capacity (left) and the elliptic capacity (right) of E = {z ∈ C : |z| ≤ r} for r = 0.5. 0 0.25 0.5 0.75 1 -16 -12 -8 [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: The relative error in the computed capacities c( [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: The domain G for computing the hyperbolic capacity (left) and the elliptic capacity (right) of E = [−r, r] × [−r, r] for r = 0.5. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]
Figure 23
Figure 23. Figure 23: The capacities caph(E) and cape(E) for E = [−r, r] × [−r, r]. For the hyperbolic capacity caph(E), the domain G is the bounded doubly connected domain exterior to the curve L and interior to the unit circle (see [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: The domain G for computing the hyperbolic capacity (left) and the elliptic capacity (right) of the closed region bordered by the amoeba-shaped boundary E. 9 Concluding Remarks Conformal invariants are important tools for complex analysis with many applications. Howeve…

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [33]

    Nasser and C

    M. Nasser and C. Green , A fast numerical method for ideal fluid flow in domains with multiple stirrers , Nonlinearity, 31 (2018), pp. 815–837

    work page 2018

  2. [13]

    Duren and R

    P. Duren and R. K ¨uhnau, Elliptic capacity and its distortion under conformal mappi ng, J. Anal. Math., 89 (2003), pp. 317–335

    work page 2003

  3. [1]

    Ahlfors , Conformal Invariants , McGraw-Hill, New York, 1973

    L. Ahlfors , Conformal Invariants , McGraw-Hill, New York, 1973

    work page 1973

  4. [2]

    G. D. Anderson, M. K. V amanamurthy, and M. Vuorinen , Conformal Invariants, Inequalities and Quasiconformal Maps , John Wiley, New York, 1997

    work page 1997

  5. [3]

    Atkinson, The Numerical Solution of Integral Equations of the Second K ind, Cambridge University Press, Cambridge, 1997

    K. Atkinson, The Numerical Solution of Integral Equations of the Second K ind, Cambridge University Press, Cambridge, 1997

    work page 1997

  6. [4]

    Austin, P

    A. Austin, P. Kravanja, and L. Trefethen , Numerical algorithms based on analytic function values at roots of unity , SIAM J. Numer. Anal., 52 (2014), pp. 1795–1821

    work page 2014

  7. [5]

    Betsakos, K

    D. Betsakos, K. Samuelsson, and M. Vuorinen , The computation of capacity of planar condensers, Publ. Inst. Math. (Beograd) (N.S.), 75 (2004), pp. 233–252

    work page 2004

  8. [6]

    Bezrodnykh, A

    S. Bezrodnykh, A. Bogatyrev, S. Goreinov, O. Grigoriev, H. H akula, and M. Vuorinen , On capacity computation for symmetric polygonal condenser s, J. Comput. Appl. Math., 361 (2019), pp. 271–282

    work page 2019

Show all 45 references
  1. [7]

    Bowman , Introduction to Elliptic Functions with Applications , English Universities Press Ltd., London, 1953

    F. Bowman , Introduction to Elliptic Functions with Applications , English Universities Press Ltd., London, 1953

    1953

  2. [8]

    Byrd and M

    P. Byrd and M. D. Friedman , Handbook of elliptic integrals for engineers and scientist s. Second edition, revised., Springer-Verlag, Berlin, 1971

    1971

  3. [9]

    Methods Funct

    D.Dautova, S.Nasyrov, and M.Vuorinen , Conformal module of the exterior of two rec- tilinear slits , Comput. Methods Funct. Theory, (2020), p. arXiv:1908.02459

    2020 arXiv

  4. [10]

    DeLillo, A

    T. DeLillo, A. Elcrat, and E. Kropf , Calculation of resistances for multiply con- nected domains using Schwarz-Christoffel transformations , Comput. Methods Funct. Theory, 11 (2011), pp. 725–745

    2011

  5. [11]

    Driscoll and L

    T. Driscoll and L. Trefethen , Schwarz-Christoffel mapping , Cambridge University Press, Cambridge, 2002

    2002

  6. [12]

    Dubinin , Condenser Capacities and Symmetrization in Geometric Func tion Theory , Springer, Basel, 2014

    V. Dubinin , Condenser Capacities and Symmetrization in Geometric Func tion Theory , Springer, Basel, 2014

    2014

  7. [14]

    Embree and L

    M. Embree and L. Trefethen , Green’s functions for multiply connected domains via con- formal mapping , SIAM Rev., 41 (1999), pp. 745–761. 25

    1999

  8. [15]

    Garnett and D

    J. Garnett and D. Marshall , Harmonic measure , Cambridge University Press, Cam- bridge, 2008

    2008

  9. [16]

    Greengard and Z

    L. Greengard and Z. Gimbutas , FMMLIB2D: A MATLAB tool- box for fast multipole method in two dimensions , version 1.2. ed., 2012. http://www.cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html. Accessed 1 Jan 2018

    2012

  10. [17]

    Hakula, A

    H. Hakula, A. Rasila, and M. Vuorinen , On moduli of rings and quadrilaterals: algo- rithms and experiments , SIAM J. Sci. Comput., 33 (2011), pp. 279–302

    2011

  11. [18]

    , Computation of exterior moduli of quadrilaterals , Electron. Trans. Numer. Anal., 40 (2013), pp. 436–451

    2013

  12. [19]

    , Conformal modulus on domains with strong singularities and cusps, Electron. Trans. Numer. Anal., 48 (2018), pp. 462–478

    2018

  13. [20]

    Hariri, R

    P. Hariri, R. Kl ´ en, and M. Vuorinen , Conformally Invariant Metrics and Quasiconfor- mal Mappings , Springer Monographs in Mathematics, Springer, Switzerland, 202 0

  14. [21]

    Keen and N

    L. Keen and N. Lakic , Hyperbolic geometry from a local viewpoint , Cambridge University Press, Cambridge, 2007

    2007

  15. [22]

    Kirsch , Transfinite diameter, Chebyshev constant and capacity , in Handbook of Complex Analysis: Geometric Function Theory, Vol

    S. Kirsch , Transfinite diameter, Chebyshev constant and capacity , in Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, R. K¨ uhnau, ed., Elsev ier, Amsterdam, 2005, pp. 243–308

    2005

  16. [23]

    Kress , A Nystr¨ om method for boundary integral equations in domain s with corners , Nu- mer

    R. Kress , A Nystr¨ om method for boundary integral equations in domain s with corners , Nu- mer. Math., 58 (1990), pp. 145–161

    1990

  17. [24]

    , Linear Integral Equations , Springer, New York, 2014

    2014

  18. [25]

    K ¨uhnau, ed

    R. K ¨uhnau, ed. , Handbook of complex analysis: geometric function theory. V ol. 1, 2002, Vol. 2, 2005 , Elsevier Science B.V., Amsterdam

    2002

  19. [26]

    Kythe , Handbook of conformal mappings and applications , Taylor & Francis Group, Boca Raton, 2019

    P. Kythe , Handbook of conformal mappings and applications , Taylor & Francis Group, Boca Raton, 2019

    2019

  20. [27]

    Liesen, O

    J. Liesen, O. S ´ ete, and M. Nasser , Fast and accurate computation of the logarithmic capacity of compact sets , Comput. Methods Funct. Theory, 17 (2017), pp. 689–713

    2017

  21. [28]

    Nasser , Numerical conformal mapping via a boundary integral equati on with the gener- alized Neumann kernel , SIAM J

    M. Nasser , Numerical conformal mapping via a boundary integral equati on with the gener- alized Neumann kernel , SIAM J. Sci. Comput., 31 (2009), pp. 1695–1715

    2009

  22. [29]

    , Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe canonical slit domains , J. Math. Anal. Appl., 382 (2011), pp. 47–56

    2011

  23. [30]

    , Fast solution of boundary integral equations with the gener alized Neumann kernel , Elec- tron. Trans. Numer. Anal., 44 (2015), pp. 189–229

    2015

  24. [31]

    , Numerical computing of preimage domains for bounded multip ly connected slit domains , J. Sci. Comput., 78 (2019), pp. 582–606

    2019

  25. [32]

    Nasser and F

    M. Nasser and F. Al-Shihri , A fast boundary integral equation method for conformal mapping of multiply connected regions , SIAM J. Sci. Comput., 33 (2013), pp. A1736–A1760

    2013

  26. [34]

    Nasser, A

    M. Nasser, A. Murid, and Z. Zamzamir , A boundary integral method for the Riemann- Hilbert problem in domains with corners , Complex Var. Elliptic Equ., 53 (2008), pp. 989–1008

    2008

  27. [35]

    Nasser and M

    M. Nasser and M. Vuorinen , Conformal invariants in simply connected domains , (2020), p. arXiv:2001.10182

    2020 arXiv

  28. [36]

    , Numerical computation of the capacity of generalized conde nsers, J. Comput. Appl. Math., 377 (2020), p. 112865. 26

    2020

  29. [37]

    Olver, D

    F. Olver, D. Lozier, R. Boisvert, and C. W. Clark, eds. , NIST Handbook of Mathe- matical Functions, Cambridge University Press, Cambridge, 2010

    2010

  30. [38]

    Papamichael and N

    N. Papamichael and N. Stylianopoulos , Numerical conformal mapping: Domain de- composition and the mapping of quadrilaterals , World Scientific, New Jersey, 2010

    2010

  31. [39]

    Saff and V

    E. Saff and V. Totik , Logarithmic potentials with external fields , Springer-Verlag, Berlin, 1997

    1997

  32. [40]

    Schinzinger and P

    R. Schinzinger and P. Laura , Conformal mapping. Methods and applications , Dover Pub- lications, Inc., New York, 2003

    2003

  33. [41]

    Trefethen and J

    L. Trefethen and J. Weideman , The exponentially convergent trapezoidal rule , SIAM Review, 56 (2014), pp. 385–458

    2014

  34. [42]

    V asil’ev, Moduli of Families of Curves for Conformal and Quasiconform al Mappings , Springer-Verlag, Berlin, 2002

    A. V asil’ev, Moduli of Families of Curves for Conformal and Quasiconform al Mappings , Springer-Verlag, Berlin, 2002

    2002

  35. [43]

    Vuorinen , Conformal invariants and quasiregular mappings , J

    M. Vuorinen , Conformal invariants and quasiregular mappings , J. Anal. Math., 45 (1985), pp. 69–115

    1985

  36. [44]

    Lecture Not es in Mathematics , Springer-Verlag, Berlin, 1988

    , Conformal geometry and quasiregular mappings. Lecture Not es in Mathematics , Springer-Verlag, Berlin, 1988

    1988

  37. [45]

    Wegmann and M

    R. Wegmann and M. Nasser , The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions , J. Comput. Appl. Math., 214 (2008), pp. 36–57. 27

    2008

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