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Numerical computation of Mityuk's function and radius for circular-radial slit domains

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A boundary integral equation with the generalized Neumann kernel computes Mityuk’s radius and function for multiply connected circular/radial slit domains, and the numerics show the boundary limits fail at radial-slit endpoints.

desk verdict Solid numerical tool for Mityuk's radius, but the slit-endpoint counterexample is numerically suggested from a ratio of two diverging quantities and the paper states it too strongly. read the letter →

arxiv 1908.03874 v2 pith:PX63YKMW submitted 2019-08-11 math.CV

El Mostafa Kalmoun , Mohamed M S Nasser , Matti Vuorinen This is my paper · ORCID
classification math.CV MSC 65E0530C3065R20
keywords MityukradiusfunctiongeneralizedreducedmodulusconformalNeumannkernelboundaryintegralequationcircularandradialslitdomainsnumericalmapping
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 develops a numerical method for Mityuk’s radius $R(G,\alpha)$ and Mityuk’s function $m(G,\alpha)$, generalizations of conformal radius and reduced modulus to multiply connected domains, when the canonical target is the unit disk with circular and/or radial slits. The method reduces the computation to a boundary integral equation with the generalized Neumann kernel, so that $R$ and $m$ are read off from a single scalar $h_0$ of the solution. The numerical experiments validate the existing theorems on critical points, including the annulus cases and the count $n_m-n_s=1-\ell$. They also produce a counterexample: when one internal boundary component is a slit mapped to a radial slit, the predicted boundary limits (6) and (7) no longer hold at the slit endpoints, where the limit is finite and depends on the direction of approach.

What carries the argument

The load-bearing object is the conformal map $\Phi_\alpha: G \to \Omega$, normalized by $\Phi_\alpha(\alpha)=0$ and $\Phi'_\alpha(\alpha)>0$, rewritten as $\Phi_\alpha(z)=c(z-\alpha)e^{(z-\alpha)f(z)}$ so that $\log R(G,\alpha)=h_0$ is the boundary constant determined by the mapping. The numerical engine is the boundary integral equation with the generalized Neumann kernel $N(s,t)=\frac{1}{\pi}\mathrm{Im}\left[\frac{A(s)}{A(t)}\frac{\eta'(t)}{\eta(t)-\eta(s)}\right]$ and its associated singular kernel $M$, solved as $(I-N)\mu=-M\gamma$ with $h=[M\mu-(I-N)\gamma]/2$. Discretization uses the trapezoidal rule with a Nyström method, graded meshes for corners and slits, and a fast iterative solver, giving $O((\ell+1)n\log n)$ cost. In the slit example, an elementary conformal map opens the slit and the transformation rule $R(G,\alpha)=R(\Psi_2(G),\Psi_2(\alpha))/|\Psi'_2(\alpha)|$ is what couples the numerical counterexample to the failure of (7).

What would settle it

Compute $R(G,\alpha)$ along the vertical and horizontal approach paths to a slit endpoint for $n=2^{12},2^{13},2^{14},2^{15},2^{16}$; if the two directional limits converge to the same value as $n$ grows, or if either sequence keeps increasing without stabilization, the claimed failure of (7) would be a numerical artifact rather than a property of Mityuk’s radius.

Watch

Extended reading notes

Core claim

The central claim is that a single boundary-integral framework computes Mityuk’s radius and Mityuk’s function for arbitrary bounded multiply connected domains whose canonical image is the unit disk with $\ell$ circular/radial slits, and that the computed values faithfully detect critical points and boundary behavior. The paper shows that the boundary value problem $\mathrm{Im}[e^{-i\theta(t)}\log\Phi_\alpha(\eta(t))]=R(t)$ for the normalized conformal map leads to the integral equation $(I-N)\mu=-M\gamma$, from whose solution one obtains $h_0=\log R(G,\alpha)$ and $m(G,\alpha)=h_0/(2\pi)$. Using this machinery, the paper confirms the theorems cited as [1, 9, 14] on critical points and boundary limits for smooth and piecewise smooth boundaries. Its main new finding is numerical: in the rectangle-with-slit example, when the canonical mapping sends the slit to a radial slit, $R(G,\alpha)$ approaches different finite values as $\alpha$ approaches a slit endpoint vertically versus along the real axis, so the limit in (7) does not exist there, contradicting a naive extension of the smooth-boundary result.

Load-bearing premise

The load-bearing premise is that the numerical values near the slit endpoints, obtained with $n=2^{15}$ and graded meshes, are accurate enough to distinguish a genuinely nonexistent limit from a large finite value, yet no convergence study or error bound is reported at those points.

Editorial extensions

If this is right

  • Mityuk’s radius and function become computable for multiply connected domains of arbitrary connectivity, with contour maps that reveal critical points and their indices.
  • The annulus results confirm the known dichotomy: infinitely many critical points on $|\alpha|=\sqrt{q}$ for a circular slit and none for a radial slit; for $\ell\ge 2$ mixed circular/radial slits, critical points exist and satisfy $n_m-n_s=1-\ell$ in the tested cases.
  • Where the boundary has corners but is not a slit (rectangle in rectangle, triangle in triangle), the limits (6) and (7) still appear to hold, so the failure is specific to the slit endpoints in the radial-slit case.
  • The observed inequality $R(G,\alpha)\ge d(\alpha,\partial G)$ holds in all tested domains and is posed as an open question for all finitely connected Jordan domains.

Reading between the lines

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

  • If the slit-endpoint failure is not a numerical artifact, Mityuk’s radius is not continuously extendable to the boundary for slit domains, so definitions of the generalized reduced modulus for such domains may need to prescribe endpoint behavior separately.
  • The direction-dependent limits suggest the endpoint asymptotics are governed by the local conformal behavior at the slit tip; a sharp asymptotic formula could be tested by computing $R$ along several directions and checking the rate of approach.
  • The inequality $R(G,\alpha)\ge d(\alpha,\partial G)$, if proven generally, would tie Mityuk’s radius to Euclidean distance and give a quantitative lower bound in conformal geometry.
  • The same boundary-integral machinery could be applied to polycircular-arc canonical domains or to countably connected domains, provided a graded-mesh convergence analysis near slit endpoints is supplied.
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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

3 major / 6 minor

Summary. The paper develops a numerical method for computing Mityuk's function and Mityuk's radius for bounded multiply connected domains, using the boundary integral equation with the generalized Neumann kernel. The canonical target domains are the unit disk with circular and/or radial slits. The method is validated against known analytic formulas for the annulus, and then applied to a series of doubly and multiply connected examples to study critical points of the radius function and the boundary limits (6) and (7). The central novel claim is that the boundary limit (7) fails when an internal boundary component is a slit mapped to a radial slit. The paper also proposes the conjectural lower bound R(G, alpha) >= d(alpha, partial G) for circular-slit canonical domains.

Significance. If the numerical method and the reported counterexample are correct, the paper provides a useful computational tool for a quantity that is central to recent work on generalized reduced moduli, and it supplies numerical evidence on open questions about critical points and boundary behavior. The method itself is built on published integral-equation solvers, so there is no circularity; the annulus validation uses independent analytic formulas; and the MATLAB code is publicly available. However, the main new claim about failure of the boundary limits rests on a single numerical experiment with no convergence study or error control near the slit endpoints, and the example as written contains a domain/mapping inconsistency that makes the transformation invalid. The significance is therefore conditional: the tool is promising, but the headline conclusion is not yet supported.

major comments (3)
  1. [§4.1.3, Eq. (17)] The domain G is defined as the rectangle minus the segment [-1,0], but the mapping Psi_1(z) = (z + 1/z)/4 + 1/2 maps the unit circle to the segment [0,1], so its inverse Psi_2 maps the exterior of [0,1] to the exterior of the unit disk and has a branch cut along [0,1]. Since the stated slit is [-1,0], the segment [0,1] lies inside G, and Psi_2 is not analytic on G. Consequently Eq. (17) is not a valid conformal transformation for this example. To map the slit [-1,0] to the unit circle one would need Psi_1(z) = (z + 1/z)/4 - 1/2, or the domain would need to be defined with slit [0,1]. This is load-bearing because the counterexample in Section 5 relies entirely on Eq. (17).
  2. [§4.1.3 and §5] The claimed failure of the limit (7) at the slit endpoints is supported only by the numerical experiment using Eq. (17), in which both R(Psi_2(G), Psi_2(alpha)) and |Psi_2'(alpha)| tend to infinity as alpha approaches an endpoint. No convergence study in n, no error estimates, and no independent asymptotic check are provided for this ratio; the statement that n = 2^15 with graded meshes is sufficient is not demonstrated. The values plotted in Figure 8 could therefore be a numerical artifact, especially since the two factors are of comparable order near the endpoints. To justify the conclusion that the limit does not exist, the authors should provide a convergence table for representative points near an endpoint (e.g., n = 2^12 through 2^18), and ideally a local asymptotic expansion of the ratio or confirmation by an independent method. As written, the statement in Section 5 that 'the limit values in (7) do not hold true' is a mathematical assertion based on unverified numerics.
  3. [§4.1.3, §4.2.2, §4.2.3] Several claims about the number and nature of critical points are inferred solely from contour plots: for example, 'eight critical points (n_m = n_s = 4)' in §4.1.3, 'twelve critical points (n_m = 4, n_s = 8)' in §4.2.2, and 'n_m - n_s = -5' in §4.2.3. The paper does not report a numerical solution of the critical-point equation phi'_alpha(alpha) = 0 (equivalently, a stationary condition for R(G, alpha)), nor residuals for the claimed critical points. Since these counts are presented as validations of theoretical results, the authors should locate the critical points numerically, for example with a zero-finding method applied to the discretized derivative or a Newton search initialized from the contour data, and report their locations and residuals.
minor comments (6)
  1. [§4.1.3] The sentence 'R(G, iy), R(G, 0.25 + iy), R(G, 0.5 + iy), and R(G, 1 + iy) for x in (-1,1)\setminus{0}' uses the variable x where y is meant; please correct this.
  2. [§4.2.3] The domain is described as 'exterior to the five circles,' but six centers are listed; correspondingly, 'five radial slits' should read 'six radial slits.'
  3. [§1] The boundary components are indexed Gamma_0, Gamma_1, ..., Gamma_m, but the connectivity is given as ell + 1; please use a consistent index set, for example Gamma_0, ..., Gamma_ell.
  4. [§4.1.3] The limit results are attributed to reference [8] in this section but to reference [9] in the Introduction; please unify the citation.
  5. [Figures 5 and 8] The text refers to 'the graph of R(G, x)' without specifying which panel of the multi-panel figures is meant; please add explicit panel references.
  6. [Figures 7-17] The term 'critical streamlines' is used but never defined; please explain how these curves are computed (for example, as level sets through saddle points or as separatrices).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solver is an external integral-equation method, the annulus case is validated against independent analytic formulas, and the slit counterexample is an explicit conformal computation rather than a fitted prediction.

full rationale

The paper's derivation chain is self-contained. Mityuk's radius is computed from the boundary integral equation with the generalized Neumann kernel, with the solver fbie attributed to the externally published reference [20] and the conformal-mapping formulation to [19]; these are computational tools, not restatements of the target quantities. The only quantity fitted to the geometry is the auxiliary boundary data, and no parameter is fitted to Mityuk's radius itself. The method is cross-checked against independent analytic series formulas for the annulus in §4.1.1, confirming both the radial-slit and circular-slit cases. The central new claim, that the boundary limits in (7) fail at slit endpoints for a radial-slit image, is obtained from the exact identity (17) R(G,α)=R(Ψ2(G),Ψ2(α))/|Ψ2'(α)|, with the elementary Joukowski-type map Ψ2 given explicitly; it is not obtained by assuming the conclusion. Any weakness about convergence of the numerical quotient at the slit endpoints is a correctness/accuracy concern, not a circularity. The self-citations to [19] and [20] are load-bearing only as numerical infrastructure and do not smuggle in the paper's conclusions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method relies on standard conformal mapping and integral equation results; no free parameters are fitted to the target quantity. All numerical parameters (n, iprec, gmrestol, maxit) are fixed computational settings, not fitted values.

assumptions (4)
  • standard math Existence and uniqueness of the conformal map from G onto the canonical circular/radial slit domain with normalization (1).
    Invoked in Section 1 to define Mityuk's radius via the mapping Φ_α; cited to Goluzin [11, Theorem 6, p. 242].
  • standard math The generalized Neumann kernel integral equation (I - N)µ = -Mγ has a unique solution for the considered domains.
    Used in Section 3 to compute the boundary values of the auxiliary function f; results from Wegmann, Murid, and Nasser [18,24,25].
  • domain assumption The Nyström method with trapezoidal rule and graded mesh converges for boundary integral equations on domains with corners.
    Used in Section 3 for piecewise smooth boundaries; cited to Kress [15] and Liesen et al. [16].
  • domain assumption The boundary limits (6) and (7) hold for smooth boundaries as proven in [8].
    Referenced in Section 4.1.3 to frame the counterexample; the paper compares its numerical results against this theoretical result.

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Pith. "Pith review of Numerical computation of Mityuk's function and radius for circular-radial slit domains." pith.science (2026). https://pith.science/paper/PX63YKMW

@misc{pith2026190803874,
  author       = {Pith},
  title        = {Pith review of: Numerical computation of Mityuk's function and radius for circular-radial slit domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PX63YKMW}},
  note         = {Machine review of arXiv:1908.03874}
}
read the original abstract

We consider Mityuk's function and radius which have been proposed in \cite{Mit} as generalizations of the reduced modulus and conformal radius to the cases of multiply connected domains. We present a numerical method to compute Mityuk's function and radius for canonical domains that consist of the unit disk with circular/radial slits. Our method is based on the boundary integral equation with the generalized Neumann kernel. Special attention is given to the validation of the theoretical results on the existence of critical points and the boundary behavior of Mityuk's radius.

Figures

Figures reproduced from arXiv: 1908.03874 by the authors.

Figure 1
Figure 1. The canonical domain Ω: The unit disk with circular/radial slits. Furthermore, given that the domain G has smooth boundary curves, it was proven in [9] that Mityuk’s radius R(G, α) is infinitely differentiable on G and has the following limit values on the boundary Γ. First, for the external boundary Γ0, we have lim α→β∈Γ0 R(G, α) = 0. (6) Then, for the internal boundary components Γk, k = 1, 2, . . . , ℓ, lim α→β∈Γ… view at source ↗
Figure 2
Figure 2. The contour maps and the surface plots of the functi [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The contour maps and the surface plots of the functi [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: The values of the radius function R(G, x) for x ∈ G with −1 < x < 1. The domain G is the same as in Figures 2 and 3. where Ψ ′ 2 (α) = 1 Ψ′ 1 (Ψ2(α)) = 4Ψ2(α) 2 Ψ2(α) 2 − 1 . Henceforth, in view of (2), Mityuk’s radius of the domain G with respect to the point α is giv…
Figure 5
Figure 5. Figure 5: The contour maps of the function R(G, α) for θ1 = π/2 (left), θ1 = 0 (center), and the values of R(G, x) for x ∈ G with −1 < x < 1 (right) where the center of the inner circle is a = 0.05 for the first row and a = 0.5 for the second row. Critical streamlines are shown …
Figure 6
Figure 6. Figure 6: The rectangle with a slit domain G (left) and its image Ψ2(G) (right) as described in §4.1.3. close to the unit circle in the domain Ψ2(G) and, by (7), the values of R(Ψ2(G), Ψ2(α)) converge to 0. Since Ψ′ 2 (α) 6= 0 even for points on the slit, then, by (17), the valu…
Figure 7
Figure 7. Figure 7: The contour maps of the function R(G, α) for θ1 = π/2 (left) and θ1 = 0 (right). Critical streamlines are shown in red color. -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 -1 -0.5 0 0.5 1 0 2 4 6 8 10 -3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: The values of R(G, iy), R(G, 0.25 + iy), R(G, 0.5 + iy), and R(G, 1 + iy), x ∈ (−1, 0) ∪ (0, 1), for θ1 = π/2 (left) and θ1 = 0 (center), and the values of R(G, x), x ∈ (−3, 0) ∪ (1, 3), for θ1 = 0 (right). The domain G is the same as in [PITH_FULL_IMAGE:figures/full_…
Figure 9
Figure 9. Figure 9: The contour maps of the function R(G, α) for θ1 = π/2 (left) and θ1 = 0 (right). Critical streamlines are shown in red color. -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 0 5 10 15 20 -3 -2 -1 0 1 2 3 0 5 10 15 20 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: The values of R(G, x), R(G, 0.25 + iy), R(G, 0.5 + iy), and R(G, 1 + iy), y ∈ (−1, 0) ∪ (0.5, 1), for θ1 = π/2 (left) and θ1 = 0 (center), and the values of R(G, x), R(x + 0.25i), and R(x + 0.5i), x ∈ (−3, 0) ∪ (1, 3), for θ1 = 0 (right). The domain G is the same as i…
Figure 11
Figure 11. Figure 11: The contour maps of the function R(G, α) for θ1 = π/2 (left) and θ1 = 0 (right). Critical streamlines are shown in red color [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: The points 3 + iy, 4 + iy, x + i, and x + 3i in the domain G (top, left) and the values of Mityuk’s radius R(G, α) at these points. (nm = 4 and ns = 8) for the case of five circular slits, and only four saddle points in the two other cases of the canonical domain. We …
Figure 13
Figure 13. Figure 13: The contour maps of the function R(G, α) for θ1 = π/2 (left) and θ1 = 0 (right). Critical streamlines are shown in red color. 5 Conclusion We have introduced a numerical method to compute Mityuk’s function and radius of multiply connected domains with respect to the c…
Figure 14
Figure 14. Figure 14: The contour maps of the function R(G, α) for: θ1 = θ2 = π/2 (left), θ1 = θ2 = 0 (center), and θ1 = π/2, θ2 = 0 (right). Critical streamlines are shown in red color. -3 -2 -1 0 1 2 3 0 0.5 1 1.5 -3 -2 -1 0 1 2 3 0 10 20 30 40 50 -3 -2 -1 0 1 2 3 0 5 10 15 20 [PITH_FUL…
Figure 15
Figure 15. Figure 15: The values of R(G, x) for −3 < x < 3 such that x ∈ G for: θ1 = θ2 = π/2 (left), θ1 = θ2 = 0 (center), and θ1 = π/2, θ2 = 0 (right). The domain G is the same as in [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: The contour maps of the function R(G, α) for: θ1 = · · · = θ5 = π/2 (first row, left), θ1 = · · · = θ5 = 0 (first row, right), θ1 = θ2 = θ3 = θ5 = π/2, θ4 = 0 (second row, left), θ1 = θ3 = θ5 = π/2, θ2 = θ4 = 0 (second row, center), and θ4 = π/2, θ1 = θ2 = θ3 = θ5 = 0…
Figure 17
Figure 17. Figure 17: The contour maps of the function R(G, α) for: θ1 = · · · = θ6 = π/2 (left), θ1 = · · · = θ6 = 0 (center), θ1 = θ2 = θ3 = π/2, θ4 = θ5 = θ6 = 0 (right). Critical streamlines are shown in red color. [7] V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric…

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Aksent’ev, M.I

    L.A. Aksent’ev, M.I. Kinder, and S.B. Sagitov a , Solvability of the exterior inverse boundary value problem in the case of a multiply conn ected domain , Trudy Sem. Kraev. Zadacham. 20 (1983) 22–34 [in Russian]

    work page 1983

  2. [2]

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

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

    work page 1997

  3. [3]

    Austin, P

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

    work page 2014

  4. [4]

    Crowdy and A

    D. Crowdy and A. Fokas , Conformal mappings to a doubly connected polycircular arc domain , Proc. R. Soc. A 463 (2007) 1885–1907

    work page 2007

  5. [5]

    Crowdy and J

    D. Crowdy and J. Marshall , Analytic formulae for the Kirchhoff–Routh path function in multiply connected domains , Proc. Roy. Soc. A. 461 (2005) 2477–2501

    work page 2005

  6. [6]

    Crowdy and J

    D. Crowdy and J. Marshall , Conformal mappings between canonical multiply connected domains, Comput. Methods Funct. Theory 6(1) (2006) 59–76. 16 Figure 17: The contour maps of the function R(G, α) for: θ1 = · · · = θ6 = π/2 (left), θ1 = · · · = θ6 = 0 (center), θ1 = θ2 = θ3 = π/2, θ4 = θ5 = θ6 = 0 (right). Critical streamlines are shown in red color

    work page 2006

  7. [7]

    V. N. Dubinin , Condenser Capacities and Symmetrization in Geometric Func tion Theory, Birkhäuser, 2014

    work page 2014

  8. [8]

    Elizarov, A.V

    A.M. Elizarov, A.V. Kazantsev, and M.I. Kinder , Strict superharmonicity of Mityuk’s function for countably connected domains of simpl e structure , Lobachevskii Journal of Mathematics 38(3) (2017) 408–413

    work page 2017

Show all 26 references
  1. [9]

    Elizarov, A.V

    A.M. Elizarov, A.V. Kazantsev, and M.I. Kinder , Generalized reduced module of a domain over the unit disc with circular and radial slits , Lobachevskii Journal of Mathematics 39(5) (2018) 664–672

    2018

  2. [10]

    Gakhov , Boundary Value Problems , Pergamon Press, Oxford, 1966

    F.D. Gakhov , Boundary Value Problems , Pergamon Press, Oxford, 1966

    1966

  3. [11]

    Goluzin , Geometric Theory of Functions of a Complex Variable , Amer

    G. Goluzin , Geometric Theory of Functions of a Complex Variable , Amer. Math. Soc., Rhode Island, 1969

    1969

  4. [12]

    Greengard and Z

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

    2018

  5. [13]

    Kazantsev , Sectio Aurea conditions for Mityuk’s radius of two-connect ed do- mains, Uch

    A.V. Kazantsev , Sectio Aurea conditions for Mityuk’s radius of two-connect ed do- mains, Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 159 (2017) 33– 46

    2017

  6. [14]

    Kinder , The number of solutions of F

    M.I. Kinder , The number of solutions of F. D. Gakhov’s equation in the case of a multiply connected domain , Sov. Math. (Iz. VUZ) 28(8) (1984) 91–95

    1984

  7. [15]

    Kress, A Nyström method for boundary integral equations in domains with corners, Numer

    R. Kress, A Nyström method for boundary integral equations in domains with corners, Numer. Math. 58(2) (1990) 145–161

    1990

  8. [16]

    Liesen, O

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

    2017

  9. [17]

    Mityuk , A generalized reduced module and some of its applications , Izv

    I.P. Mityuk , A generalized reduced module and some of its applications , Izv. Vyssh. Uchebn. Zaved., Mat. 2 (1964) 110–119 [in Russian]

    1964

  10. [18]

    Murid and M.M.S

    A.H.M. Murid and M.M.S. Nasser , Eigenproblem of the generalized Neumann kernel, Bull. Malays. Math. Sci. Soc. 26 (2003) 13–33

    2003

  11. [19]

    Nasser , Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonica l slit domains , J

    M.M.S. Nasser , Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonica l slit domains , J. Math. Anal. Appl. 382 (2011) 47–56. 17

    2011

  12. [20]

    Nasser , Fast solution of boundary integral equations with the gener alized Neumann kernel , Electron

    M.M.S. Nasser , Fast solution of boundary integral equations with the gener alized Neumann kernel , Electron. Trans. Numer. Anal. 44 (2015) 189–229

    2015

  13. [21]

    Nasser, T

    M.M.S. Nasser, T. Sakajo, A.H.M. Murid and L.K. Wei , A fast computational method for potential flows in multiply connected coastal dom ains, Japan J. Indust. Appl. Math. 32 (2015) 205–236

    2015

  14. [22]

    Trefethen and J

    L.N. Trefethen and J. A. C. Weideman , The exponentially convergent trape- zoidal rule , SIAM Review 56 (2014) 385–458

    2014

  15. [23]

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

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

    2002

  16. [24]

    Wegmann, A.H.M

    R. Wegmann, A.H.M. Murid, and M.M.S. Nasser , The Riemann-Hilbert prob- lem and the generalized Neumann kernel , J. Comput. Appl. Math. 182 (2005) 388–415

    2005

  17. [25]

    Wegmann and M.M.S

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

    2008

  18. [26]

    Wen , Conformal Mapping and Boundary Value Problems , Amer

    G.C. Wen , Conformal Mapping and Boundary Value Problems , Amer. Math. Soc., Providence, 1992. 18

    1992

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