Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse

Abdelhamid Rehouma

REVIEW 3 major objections 4 minor 20 references

Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Gronwall's area formula yields a closed-form area for m-leaf lemniscates and an orthonormal system of Chebyshev polynomials on the ellipse.

desk verdict Textbook re-derivations with a false orthogonality theorem as stated; not a research contribution, but the lemniscate area computation is correct and clean. read the letter →

arxiv 2506.00663 v1 pith:ES6DMXXK submitted 2025-05-31 math.CV

Abdelhamid Rehouma This is my paper

classification math.CV MSC 42C0533C45

keywords Gronwall'sareaformulaconformalmappinglemniscateTchebychevpolynomialsofthesecondkindellipseorthogonalityintegralinterpolationunivalentfunctions

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 uses Gronwall's area formula, which expresses the area enclosed by the image of a circle under a conformal map as a sum over Laurent coefficients, to compute areas of circles, ellipses, and m-leaf symmetric lemniscates. It derives a closed-form Gamma-function expression for the area of the lemniscate $\{|w^m-1|=1\}$ and shows that the same coefficient-counting method recovers the classical circle and ellipse areas. In the second part, it proves that suitably normalized second-kind Chebyshev polynomials form an orthonormal system for the area integral over the filled ellipse with foci at $\pm 1$, and it gives the squared norms of $T_n'$ in closed form. It closes with an interpolation result on the ellipse whose error decays geometrically on the focal interval $[-1,1]$.

What carries the argument

The central object is Gronwall's area formula: if $\Psi_r(z)=z+\sum_{n=0}^\infty b_n/z^n$ conformally maps the exterior of $|z|=r$ onto the exterior of a curve, then the enclosed area is $A_r=\pi(r^2-\sum_{n=1}^\infty n|b_n|^2r^{-2n})$. The argument proceeds by expanding a given exterior map in a Laurent series, reading off the coefficients, and summing their squared moduli; for the lemniscate, the binomial expansion of $(1+1/z^m)^{1/m}$ supplies the coefficients, and the binomial-coefficient sums together with the gamma duplication formula turn the series into a closed form. For the ellipse, the carrying mechanism is the conformal pullback $z=\cos w$, whose Jacobian $|1-z^2|$ converts the area integral over the ellipse into a trigonometric double integral over a rectangle, where sine orthogonality produces the normalization constants.

What would settle it

Take $m=3$ and observe whether $\Psi(z)=(z^3+1)^{1/3}$ is injective on $|z|>1$: since $\Psi(e^{2\pi i/3}z)=\Psi(z)$, it is not, so the Gronwall-coefficient route cannot be valid as stated; independently, integrate the polar equation $\rho^3=2\cos 3\phi$ on the three leaves and compare with Eq. (36) to test the area formula itself.

Watch

Extended reading notes

Core claim

Two theorems carry the paper. Theorem 7 states that applying Gronwall's area formula, $A_r=\pi(r^2-\sum_{n=1}^\infty n|b_n|^2r^{-2n})$, to the exterior map $\Psi(z)=z(1+1/z^m)^{1/m}$ gives the area of the $m$-leaf symmetric lemniscate $\{w:|w^m-1|=1\}$ as $A=2^{2/m-1}\Gamma(1/m+1/2)/(\Gamma(1/m+1))\sqrt{\pi}$, while the same sum yields $\pi r^2$ for a circle and $\pi ab$ for an ellipse. Theorem 11 states that on the filled ellipse with foci at $\pm 1$, parameterized by $z=\cos w$ with $a=\cosh c$, $b=\sinh c$, and $\rho=a+b$, the polynomials $P_n(z)=2\sqrt{(n+1)/\pi}(\rho^{n+1}-\rho^{-n-1})^{-1/2}U_n(z)$ satisfy $\int\!\int_D P_n(z)P_m(z)\,dxdy=\delta_{n,m}$, and $\int\!\int_D |T_n'(z)|^2\,dxdy=n\pi(\rho^n-\rho^{-n})/4$. The paper also proves that Lagrange interpolation at the roots of $U_n$ converges on $[-1,1]$ with error $O(R^{-n})$ for functions analytic in the surrounding ellipse.

Load-bearing premise

The area-formula proof for lemniscates assumes the unproved premise that $\Psi(z)=z(1+1/z^m)^{1/m}$ is a one-to-one conformal map from the exterior of the unit disk to the exterior of the lemniscate for the chosen branch; if this fails, Gronwall's coefficient formula cannot be applied.

Editorial extensions

If this is right

The area of the $m$-leaf lemniscate $\{|w^m-1|=1\}$ is fixed by the Gamma-function expression in Theorem 7, removing the need for numerical quadrature for these areas.
The same Gronwall coefficient counting recovers the classical circle and ellipse areas, $\pi r^2$ and $\pi ab$, showing the method as a unified area tool.
The normalized polynomials $P_n$ are orthonormal for the area integral over the filled ellipse, with the normalization constant determined solely by $\rho=a+b$.
The squared norm of $T_n'$ over the ellipse has the closed form $n\pi(\rho^n-\rho^{-n})/4$.
Interpolation at the roots of $U_n$ converges geometrically on $[-1,1]$ for analytic functions in the surrounding ellipse, with error $O(R^{-n})$.

Reading between the lines

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

A branch-sensitive version of Theorem 7 is worth testing: if the exterior map is restricted to a sector of angle $2\pi/m$, univalence may hold, and the same Laurent coefficients would give the area of one leaf, with the total area fixed by $m$-fold symmetry.
The rectangle-pullback proof on the ellipse suggests a general recipe: any domain that is the conformal image of a rectangle with sine-type orthogonal functions will yield explicit orthonormal polynomials; the disk identities in Proposition 8, $A=\frac{1}{4\pi}\mathfrak{I}(f)$, point to analogous coefficient-counting formulas for other domains.
The interpolation bound $O(R^{-n})$ on $[-1,1]$ is the analogue of classical polynomial approximation estimates; the same coefficient-counting framework could be used for Lebesgue-function bounds or Faber-series convergence on lemniscates, which the paper does not address.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: nothing here is new, and the central statements as printed are partly wrong. The usable core is the derivation of Gronwall's area formula and the m-leaf lemniscate area; the orthogonality theorem needs a conjugate bar before it becomes true.

What the paper does well: the proof of Gronwall's theorem (Theorem 6) is readable, and the polar-coordinate/Beta-function verification of A = 2^{2/m-1} ... is a clean, correct computation. The Joukowski map for the ellipse is standard but correctly applied. If the author explicitly chose the branch of (1+1/z^m)^{1/m} that is univalent outside the disk, Theorem 7 would be fine.

Soft spots, in order of severity. Theorem 11 is false as stated. Equation (59) writes ∫∫_D P_n(z)P_m(z) dxdy = δ_nm without conjugation, and the proof secretly computes with sin(nw)sin(m\bar w). A direct n=0, m=4 evaluation on the ellipse gives a nonzero value. Put the conjugate on P_m and you recover a classical Szego/Smirnov orthogonality; without it, the theorem collapses. Proposition 10 reports finite values for integrals whose integrands have a non-integrable |z-zp|^{-2} singularity at an interior point; those integrals diverge. Equation (33) defines the lemniscate boundary incorrectly as |w^m−1|=0; it should be =1. The prose is typo-heavy, several steps are skipped, and the single self-citation [19] is unrelated to the content.

The paper would be useful as an expository worksheet for a complex analysis course, but as a research preprint it has no new organizing principle and its headline result is wrong as written. I would not cite it, and I would not send it to referees; a desk reject with an invitation to resubmit a corrected expository note is the appropriate move.

Referee Report

3 major / 4 minor

Summary. The manuscript applies Gronwall's area formula to exterior conformal mappings to compute the areas of circles, ellipses, and m-leaf lemniscates, and then studies orthonormal systems of Chebyshev polynomials of the second kind on an ellipse, closing with an interpolation result. The two headline claims are Theorem 7, which gives a closed-form area formula for an m-leaf symmetric lemniscate, and Theorem 11, which asserts that the polynomials P_n form an orthonormal system on the ellipse. The paper also contains auxiliary results on area integrals in Propositions 8 and 10.

Significance. If the main results were correct, Theorem 7 and Theorem 11 would be useful additions to the classical literature: the lemniscate area formula is closed-form and the orthonormal system is explicit. The polar-coordinate verification inside Theorem 7 is a genuine independent check of the final area formula, and the normalization constants in Theorem 11 are plausible once the intended inner product is Hermitian. However, the manuscript as written contains a false statement in Theorem 11, a false proposition in Proposition 10, and an incomplete proof of Theorem 7, so these strengths do not support publication in the current form.

major comments (3)

[Section 4, Eq. (59)] The stated identity ∫∫_D P_n(z)P_m(z) dxdy = δ_{n,m} is false because the integral is not Hermitian. The proof at Eq. (62) and the following evaluation uses sin(mu - imv) = sin(m \bar w), i.e. the conjugate factor, so the computation establishes orthogonality for ∫∫_D P_n(z) \overline{P_m(z)} dxdy, not for the integral as written. A direct check with n=0 and m=4 gives ∫∫_D U_0(z)U_4(z) dxdy = 2π(c - ab) ≠ 0 for c>0, so the stated equality fails even before the normalization constants are inserted. The theorem should be restated with \overline{P_m(z)} in the integrand.
[Section 3, Proposition 10, Eqs. (52)-(57)] The integrands |z-z_p|^{-2} have non-integrable singularities at the interior point z_p, so the integrals I_p and J_p diverge. The proof's substitution w = z_p/r places a pole on the integration contour when r = |z_p|, and the subsequent integral ∫_0^1 2r/(r^2-|z_p|^2) dr diverges logarithmically. Consequently Eqs. (54)-(57) cannot hold as stated.
[Section 3, Theorem 7, Eqs. (34)-(37)] The proof applies Gronwall's area formula (24) to Ψ(z)=z(1+z^{-m})^{1/m} without specifying the branch or proving that the chosen branch is univalent on the exterior of the unit disk. For odd m the principal branch has branch cuts that can intersect the unit circle; if the map is not univalent there, Eq. (35) does not follow from the cited area theorem. The subsequent polar-coordinate computation independently proves (36), but the Gronwall derivation as written is incomplete.

minor comments (4)

[Section 3, Eq. (33)] The boundary of an m-leaf symmetric lemniscate is {w ∈ C : |w^m - 1| = 1}, not {w ∈ C : |w^m - 1| = 0}; the latter is just the set of m roots of unity. The subsequent text uses the correct condition, so this is a definitional typo, but it should be fixed.
[Section 2, Eq. (26)] In the proof of Theorem 6, the displayed identity should involve the conjugate factor, i.e. Ar = Im(1/2 ∫ \overline{Ψ_r(z)} Ψ'_r(z) dz); the computation that follows uses (u-iv), which is the conjugate of Ψ_r. The equation as written omits the conjugation.
[Section 4, Theorem 11] The symbol ρ in Eq. (59) is not defined in the statement of the theorem; it is only defined later in Eq. (67) as (a+b)^2. The statement should define ρ or explicitly refer to Eq. (67).
[Section 5, Proposition 12] The convergence claim in Eq. (77) depends on an implicit assumption that f is analytic in a sufficiently large ellipse E_R with R>1; the statement should include this hypothesis explicitly to make the error estimate O(1/R^n) meaningful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: all central results are derived from standard external theorems and identities, with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's derivations are self-contained in the sense relevant to circularity. Section 2 proves Gronwall's area formula from Green's theorem and standard complex integration, then applies it to the explicit conformal maps z, z + 1/z, and z(1 + 1/z^m)^{1/m} for circle, ellipse, and lemniscate. Theorem 7's area formula (Eq. 36) is obtained by expanding the explicit map via the binomial series and summing the resulting coefficient series using classical Gamma-function identities, and it is independently confirmed by direct polar-coordinate integration of the lemniscate equation. The orthogonality discussion in Theorem 11 similarly starts from the explicit conformal map z = cos w and standard trigonometric integral evaluations; no parameter is fitted to any dataset and no 'prediction' is a repackaged input. The single self-citation, [19], appears only in the reference list and is never invoked in any proof or assumption. The paper does contain serious mathematical issues: the lemniscate map's branch and univalence are not justified, and Theorem 11 omits complex conjugation in the inner product, so the stated relation (59) is false as written. These are correctness errors, not circularity: the derivations do not reduce to their own conclusions, and no claim is true merely by definition or by citation. The reader's assessment of circularity score 0.0 is therefore consistent with the evidentiary record.

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

The paper introduces no empirical parameters or invented entities. It relies on standard theorems (Caratheodory, Green, residue calculus), on two quoted binomial/Gamma series identities from Gradshteyn and Ryzhik, and on two domain-specific conformal maps whose branches and univalence are not fully proved.

assumptions (5)

standard math Caratheodory's theorem on boundary values of conformal maps (Theorem 2).
Invoked to justify the area formula for conformally mapped regions.
standard math Green's theorem and polar-coordinate area integrals (Eqs. (8) and (22)).
Used to derive Gronwall's area formula and the area integrals in Sections 2 and 3.
domain assumption The map z=cos w maps the rectangle R onto the ellipse slit along [-a,-1] and [1,a] with Jacobian |1-z^2| (Eq. 61).
Unproved in the paper but standard; it is load-bearing for the Chebyshev orthogonality derivation.
ad hoc to paper At infinity, Psi(z)=z(1+1/z^m)^{1/m}=z sum C^alpha_n z^{-mn}, and the chosen branch is univalent on the exterior of the unit disk (Eqs. (34) and (37)).
The paper does not specify the branch or prove univalence; this is a load-bearing premise for Theorem 7.
standard math The series identities sum (C^alpha_k)^2 = Gamma(2 alpha+1)/(Gamma(alpha+1))^2 and sum k (C^alpha_k)^2 = Gamma(2 alpha)/(Gamma(alpha))^2 (Eqs. (38) and (39)), together with the Gamma duplication formula.
Quoted from Gradshteyn and Ryzhik and used to derive Eq. (36).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse." pith.science (2026). https://pith.science/paper/ES6DMXXK

@misc{pith2026250600663,
  author       = {Pith},
  title        = {Pith review of: Counting methods of area integrals and Tchebychev polynomials of second kind on the ellipse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ES6DMXXK}},
  note         = {Machine review of arXiv:2506.00663}
}

read the original abstract

We use Gronwall's area formula to find the area of some differents regions as circles, ellipses and lemniscates.We use Laurent and Taylor series expansions of conformal mapping from the exterior of the unit disk to either of these regions to compute the area of them.We close this work with the discussion of orthogonal Tchebychev polynomials of second kind on the ellipse and interpolation.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

[19]

Algerian Journal of Engineering Architec- tureand Urbanism Vol

Abdelhamid REHOUMA , Asymptotic of polar orthogonal polynomials on the unit circle and their reversed polynomials. Algerian Journal of Engineering Architec- tureand Urbanism Vol. 5 Nr. 2, 2021

work page 2021
[1]

Szego , Orthogonal Polynomials, 4th ed., American Mathematical Society Collo- quium Publications, Vol.23, Amer

G. Szego , Orthogonal Polynomials, 4th ed., American Mathematical Society Collo- quium Publications, Vol.23, Amer. Math. Soc. Providence,RI, 1978

work page 1978
[2]

L V.Ahlfors,Complex Analysis ,3 rd ed ,McGraw-Hill ,New York ,1979

work page 1979
[3]

JH.Curtiss, Faber polynomials and the Faber series ,Amer .Math .Monthly 78 (1971),pp 577-596

work page 1971
[4]

WR.Derrick,Comlpex analysis and Applications ,2nd ed.,Books /Cole Pub Co .,Pacine Grove, CA 1972

work page 1972
[5]

Abramowitz and I.A

M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, 10th Edition, Dover, New York, 1972

1972
[6]

I. S. Gradshteyn, I. M. Ryzhik , Table of Integrals, Series and Products, Seventh edition. Translated from Russian by Scripta Technica, Inc.Academic Press.2007

work page 2007
[7]

Gaier, Konstruktive Methoden der konformen Abbildung, SpringerVerlag, Berlin, 1964

D. Gaier, Konstruktive Methoden der konformen Abbildung, SpringerVerlag, Berlin, 1964

work page 1964

Show all 20 references

[8]

————, Polynomial approximation of conformal maps, Constr. Approx. 14 (1998), 27-40

1998
[9]

M. V. Keldysh and M. A. Lavrentiev , On the theory of conformal mappings, Dokl. Akad. Nauk SSSR 1 (1935), 85-87. (Russian)

1935
[10]

M. V. Keldysh and M. A. Lavrentiev , Sur la representation conforme des do- maines limit´ es par des courbes rectifiables, Ann. Sci. Ecole ´ Norm.Sup. 54 (1937), 1-38

1937
[11]

Y A.L.Gueronimus,Sur des polynˆ omes extr´ emaux dans l’espace L2 (σ) .En russe .Mat .Sbornik.31 (73) 1952 , No 4 , 3-26. 20

1952
[12]

Julia, Lecons sur la repr´ esentation conforme des aires simplement connexes, Paris, 1931

G. Julia, Lecons sur la repr´ esentation conforme des aires simplement connexes, Paris, 1931

1931
[13]

40, Cambridge University Press, Cambridge, 1980

P .Koosis,Introduction to H p spaces.London Math.Soc.Lecture Notes,Series, Vol. 40, Cambridge University Press, Cambridge, 1980

1980
[14]

E.I.Pritsker,On the local asymptotics of Faber polynomials .Proc.Ame.Math.Soc ,127 (10) (1999), 2953-2960

1999
[15]

d’Analyse Math

———————,Convergence of Julia polynomials, J. d’Analyse Math. 94 (2004), 343-361

2004
[16]

V.J.Smirnov,Sur la th´ eorie Th´ eorie des polynˆ omes orthogonaux ` a une variable com- plexe.Journ.Soc.Phys.Math.de Leningrad,2 (1928) 155-179.M

1928
[17]

V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable:Constructive Theory, MIT Press, Cambridge, 1968

1968
[18]

New York .1975

N.Zeev, Conformal mapping ,Dover publications , INC. New York .1975

1975
[20]

Dilcher, K

K. Dilcher, K. B. Stolarsky , Resultants and Discriminants of Chebyshev and related polynomials, Transactions of the Amer. Math. Soc. 357 (2004), 965-981. 21

2004

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

Algerian Journal of Engineering Architec- tureand Urbanism Vol

Abdelhamid REHOUMA , Asymptotic of polar orthogonal polynomials on the unit circle and their reversed polynomials. Algerian Journal of Engineering Architec- tureand Urbanism Vol. 5 Nr. 2, 2021

work page 2021

[2] [1]

Szego , Orthogonal Polynomials, 4th ed., American Mathematical Society Collo- quium Publications, Vol.23, Amer

G. Szego , Orthogonal Polynomials, 4th ed., American Mathematical Society Collo- quium Publications, Vol.23, Amer. Math. Soc. Providence,RI, 1978

work page 1978

[3] [2]

L V.Ahlfors,Complex Analysis ,3 rd ed ,McGraw-Hill ,New York ,1979

work page 1979

[4] [3]

JH.Curtiss, Faber polynomials and the Faber series ,Amer .Math .Monthly 78 (1971),pp 577-596

work page 1971

[5] [4]

WR.Derrick,Comlpex analysis and Applications ,2nd ed.,Books /Cole Pub Co .,Pacine Grove, CA 1972

work page 1972

[6] [5]

Abramowitz and I.A

M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, 10th Edition, Dover, New York, 1972

1972

[7] [6]

I. S. Gradshteyn, I. M. Ryzhik , Table of Integrals, Series and Products, Seventh edition. Translated from Russian by Scripta Technica, Inc.Academic Press.2007

work page 2007

[8] [7]

Gaier, Konstruktive Methoden der konformen Abbildung, SpringerVerlag, Berlin, 1964

D. Gaier, Konstruktive Methoden der konformen Abbildung, SpringerVerlag, Berlin, 1964

work page 1964