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arxiv: 2606.28579 · v1 · pith:WB55PN2Mnew · submitted 2026-06-26 · 🌊 nlin.SI · math-ph· math.CA· math.MP

Transition asymptotics for the real solutions of the sinh-Gordon Painlev\'e III equation

Kenta Miyahara , Maxim L. Yattselev This is my paper

Pith reviewed 2026-06-30 00:39 UTC · model grok-4.3

classification 🌊 nlin.SI math-phmath.CAmath.MP
keywords Painlevé IIIsinh-Gordon equationasymptoticstransition asymptoticsmonodromy parameterelliptic functionsreal solutionssingular and smooth solutions
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The pith

The asymptotics of real solutions to the sinh-Gordon Painlevé III equation transition between exponential, elliptic, and trigonometric regimes depending on the scaling parameter ϰ.

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

This paper studies real-valued solutions on the positive real line of the sinh-Gordon Painlevé III equation. These solutions are parametrized by a complex monodromy parameter p with modulus greater than one, or by an additional real parameter when p is infinite. Building on prior work that gave large-x asymptotics for fixed p, the authors examine the regime where both x and p tend to infinity while satisfying 2 Im(p) equals minus s to the R. They introduce the scaling |p| squared equals one plus e to the 2ϰx and track how the leading asymptotics evolve with ϰ. A reader would care because the result supplies a unified description that continuously connects the previously separate cases of singular and smooth solutions.

Core claim

If we parametrize |p|^2 = 1 + e^{2ϰ x}, then the smooth exponential asymptotics of the solutions extends to the region ϰ>1, with a change of the leading order term at ϰ=2; at ϰ=1 the exponential behavior transitions into an elliptic asymptotics, which holds for all 0<ϰ<1; as ϰ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all p fixed.

What carries the argument

The scaling relation |p|^2 = 1 + e^{2ϰ x} combined with the condition 2 Im(p) = -s^R that interpolates between finite and infinite p while maintaining reality of the solution.

If this is right

  • Exponential asymptotics persist for ϰ greater than 1, but the leading term changes when ϰ exceeds 2.
  • Elliptic asymptotics govern the behavior throughout the interval 0 < ϰ < 1.
  • Trigonometric asymptotics emerge in the limit of small ϰ and remain valid for any fixed p.
  • The transition point ϰ equals 1 marks the boundary between exponential and elliptic regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling might be used to derive higher-order corrections that are uniform across the transition.
  • These transition results could inform the study of other Painlevé equations that admit similar monodromy parametrizations.
  • Numerical integration of the differential equation for intermediate ϰ values could confirm the predicted elliptic behavior.

Load-bearing premise

The specific scaling |p|^2 = 1 + e^{2ϰ x} together with the relation 2 Im(p) = -s^R correctly captures the continuous transition between the singular solutions (|p| finite) and the smooth solutions (p = ∞) while preserving the reality condition on (0, ∞).

What would settle it

Numerical computation of a solution for ϰ slightly larger than 1 and slightly smaller than 1, checking whether the large-x behavior switches from exponential decay or growth to oscillatory elliptic type at exactly ϰ=1.

Figures

Figures reproduced from arXiv: 2606.28579 by Kenta Miyahara, Maxim L. Yattselev.

Figure 1
Figure 1. Figure 1: The type of behavior of 𝑢p𝑥; 𝑝p𝜘, 𝑥qq for each choice of p𝜘, 𝑥q [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphs of 𝑦 “ sdp𝑥, 𝑘q for 𝑘 “ 0, 0.5, 0.8, 1; 𝑦 “ sdp𝑥, 0q “ sinp𝑥q (blue solid curve); 𝑦 “ sdp𝑥, 0.5q (purple dashed curve); 𝑦 “ sdp𝑥, 0.8q (pink dash-dotted curve); 𝑦 “ sdp𝑥, 1q “ sinhp𝑥q (red long￾dash-dotted curve). where 𝐾p¨q and 𝐸p¨q are the complete elliptic integrals of the first and second kind: (1.0.16) 𝐾p𝑘q “ ż 1 0 1 a p1 ´ 𝑦 2qp1 ´ 𝑘 2𝑦 2q 𝑑𝑦 and 𝐸p𝑘q “ ż 1 0 d 1 ´ 𝑘 2𝑦 2 1 ´ 𝑦 2 𝑑𝑦. Recall al… view at source ↗
Figure 3
Figure 3. Figure 3: The contour Γ𝜌 and the jump matrices 𝐺Ψˆ for RHP 1. Riemann-Hilbert Problem 1. Find a 2 ˆ 2 matrix function Ψˆ p𝜆q such that (1) Ψˆ p𝜆q is analytic in CzΓ𝜌, where Γ𝜌 “ iR Y 𝑆 𝜌 is oriented as on [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The jump matrices 𝐺𝑌 p𝜆q for RHP 2. Therefore, we can conclude that |𝐴 ´ 1| ď 2|𝑠 R |`𝑒 ´𝜘𝑥 (2.2.4) . Thus, the jump 𝐺𝑌 p𝜆q is close 𝐼 when (1.0.6) holds. We make this statement more precise in the next subsection. 2.3 Asymptotic Analysis Recall that 𝜘1 “ mint1, 𝜘 ´ 1u. Throughout the paper, }𝐴} 2 𝐿2pΓq :“ ż Γ }𝐴p𝜆q}2 |𝑑𝜆| and }𝐴}𝐿8pΓq :“ ess sup Γ }𝐴p𝜆q}, where }𝐴} is the Frobenius norm of a matrix 𝐴. We … view at source ↗
Figure 5
Figure 5. Figure 5: The contour Γˆ and the jump matrices 𝐺𝑌ˆp𝜆q. Riemann-Hilbert Problem 3. Find a 2 ˆ 2 matrix function 𝑌ˆp𝜆q such that (1) 𝑌ˆp𝜆q is analytic in CzΓˆ, where contour Γˆ is depicted on [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The contour Γ𝑍 and the jump matrices 𝐺𝑍p𝜆q. Note that 𝐺𝑌ˆp𝜆q is stated on [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The contour Γ𝑍 and the jump matrices 𝐺𝑍ˆ𝑛 p𝜆q for RHP 11. Note that 𝐺𝑌ˆp𝜆q is stated on [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The branch cut for a 𝑓p𝜇q and its signs on the axes. (5.1.4) b 𝑓p𝜇q “ 𝜇 2 ` 1 ´ Rep𝑎 2 q𝜇 ´2 ` O ` 𝜇 ´4 ˘˘ , 𝜇 Ñ 8 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The jump matrices 𝐺𝑋p𝜆q for RHP 12. where we used (5.1.15) as well as (2.2.4) to get the last equality. Observe that the diagonal entries are also exponentially close to zero away from 𝑎, 𝑎 since |Πp𝜆q| ă 𝜘 by (5.1.14) and Lemma 5.1. Similarly, it holds on Γp´𝑎, ´𝑎q that 𝐺𝑋p𝜆q “ ˆ ´𝑒 ´𝑥p𝜘`Πp𝜆qq ´𝐴𝑒´2𝜋i𝑥𝑉 𝐴𝑒2𝜋i𝑥𝑉 𝑒 ´𝑥p𝜘´Πp𝜆qq˙ “ ˆ ´𝑒 ´𝑥p𝜘`Πp𝜆qq ´𝑒 ´2𝜋i𝑥𝑉 ` Op|𝑠 R|`𝑒 ´𝜘𝑥q 𝑒 2𝜋i𝑥𝑉 ` Op|𝑠 R|`𝑒 ´𝜘𝑥q 𝑒 ´𝑥p𝜘´Πp𝜆q… view at source ↗
Figure 10
Figure 10. Figure 10: The jump 𝐺𝑋˜p𝜆q and the contour Γ˜ for RHP 13. (3) it holds that 𝑋˜p𝜆q “ # 𝐼 ` Op1{𝜆q as 𝜆 Ñ 8, 𝑒 𝑥ℓ 𝜎3𝑃0𝜎1𝜎3𝑒 ´𝑥ℓ 𝜎3 p𝐼 ` Op𝜆qq as 𝜆 Ñ 0. One can readily see from Lemma 5.3 that the jump matrix 𝐺𝑋˜p𝜆q is exponentially close to the identity on the arcs 𝛾 𝑖𝑛 𝑖 , 𝛾𝑜𝑢𝑡 𝑖 away from their endpoints. 5.3 Global Parametrix If we replace 𝐺𝑋˜p𝜆q by 𝐼 everywhere it is close to 𝐼 asymptotically as 𝑥 Ñ 8, we shall ar… view at source ↗
Figure 11
Figure 11. Figure 11: Homology basis for 𝔖 depicted on 𝔖p0q (which is identified with CzΔ). We fix a homology basis, 𝜶-cycle and 𝜷-cycle, on 𝔖 as indicated in [PITH_FULL_IMAGE:figures/full_fig_p043_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Planar model of 𝔖 and the 𝜶- and 𝜷-cycles. 5.3.2 Abel-Jacobi map Recall that the space of holomorphic differentials on any elliptic curve is one-dimensional, and we denote the unique (up to normalization) holomorphic differential on 𝔖 by 𝜔 “ ¨ ˝ ¿ 𝜶 𝑑𝑧 𝑤p𝒛q ˛ ‚ ´1 𝑑𝑧 𝑤p𝒛q (5.3.2) with normalized periods ¿ 𝜶 𝜔 “ 1 and ¿ 𝜷 (5.3.3) 𝜔 “: 𝜏. According to our choice of 𝜶 and the normalization of 𝑤, we have that… view at source ↗
Figure 13
Figure 13. Figure 13: Image of 𝔖𝜶,𝜷 under the map a. The black labels Γp´𝑎,¯ 𝑎¯q˘ and 𝔖p0,1q denote the domain of a. To close this subsection, we show that ap𝔖𝜶,𝜷q “ ␣ 𝑧 | 2| Rep𝑧q| ă 1, 2| Imp𝑧q| ă |𝜏| ( , [PITH_FULL_IMAGE:figures/full_fig_p045_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The jump matrices and the jump contour for ΦAip𝜁q. 5.4.2 Conformal Map To use Airy parametrix from the previous subsection, we need to introduce a conformal map of a neighborhood of 𝑎 (resp. 𝑎, ´𝑎, ´𝑎) to the 𝜁-plane where ΦAip𝜁q is defined. In this section, we provide a detailed construction around 𝑎, understanding that the other points can be handled similarly. Let Q1 :“ t𝜆 | Rep𝜆q, Imp𝜆q ą 0u be the fi… view at source ↗
Figure 15
Figure 15. Figure 15: The interior of the unit disk in the first quadrant (left panel) and its image under 3 2 𝑥ℎ1p𝜆q (right panel) [PITH_FULL_IMAGE:figures/full_fig_p052_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Images of Q1 X t|𝜆| ă 1u under the maps p 3 2 𝑥ℎ1p𝜆qq 1 3 (left panel) and ` p 3 2 𝑥ℎ1p𝜆qq 1 3 ˘2 (right panel). (5.4.7) 𝜁1p𝜆q :“ ` p 3 2 𝑥ℎ1p𝜆qq 1 3 ˘2 , which maps Q1 X t|𝜆| ă 1u conformally into the shaded region depicted on the right panel of [PITH_FULL_IMAGE:figures/full_fig_p053_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The vertical rays ℓ1, ℓ2 and the distance |𝜁1p𝜆q|, 𝜆 P BD1, which is always greater than p 3 4 𝜋𝑉𝑥q 2 3 in the depicted case. for 𝜆 P D1z𝜁 ´1 1 ` t|𝑧| ă 𝑅˚u ˘ , where Op¨q is independent of 𝜘, 𝑥, and 𝛿. Moreover, 𝑁p𝜆q “ O ˆ 1 𝛿K˚ 𝛼 ˙ for 𝜆 P Q1zD1 and p𝜘, 𝑥q R Z𝛿, where again Op¨q is independent of 𝜘, 𝑥, and 𝛿. Proof. It follows from the first claim of Lemma 5.7 that 𝜁1pD1q contains a disk of radius 𝑅p𝜘, … view at source ↗
Figure 18
Figure 18. Figure 18: The contour Γ𝑅 for RHP 18. In fact, one can easily see from Lemma 5.3 and the form of 𝐺𝑋˜p𝜆q on the imaginary axis that the above estimate remains valid on iRzBD. In particular, we have that (5.5.4) }𝐺𝑅 ´ 𝐼}𝐿2piRzBDqX𝐿8piRzBDq “ O ` |𝑠 R |𝛿 ´2 pK ˚ 𝛼 q ´2 𝑒 ´𝜘𝑥˘ (this is an estimate on the green parts of the contour on [PITH_FULL_IMAGE:figures/full_fig_p063_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The jump matrices 𝐺𝑊p𝜆q for RHP 20. diagonal matrix 𝑒 𝜘𝑥 𝜎3 in the formulae below, where one needs to recall that 𝐵 R “ 𝑒 ´𝜘𝑥 . More precisely, we have that 𝐺𝑊p𝜆q “ ˆ ´𝐵 R ´𝐴𝑒2𝑥 𝜑p𝜆q 𝐴𝑒´2𝑥 𝜑p𝜆q 𝐵 R ˙ “ ˆ 1 ´𝐴𝑒𝑥p𝜘`2𝜑p𝜆qq 0 1 ˙ ˆ 1 0 𝐴𝑒´𝑥p𝜘`2𝜑p𝜆qq 1 ˙ 𝑒 𝜘𝑥 𝜎3 “: 𝑆 ˚ 𝐿1 p𝜆q𝑆 ˚ 𝑅1 p𝜆q𝑒 𝜘𝑥 𝜎3 (6.1.1) for 𝜆 P 𝑆 1 X Q1, where, as before, Q𝑖 is the 𝑖-th quadrant. In a similar way we get that 𝐺𝑊p𝜆q “ ˆ ´𝐵 R ´𝐴𝑒2𝑥 … view at source ↗
Figure 20
Figure 20. Figure 20: The zero level curve Rep2𝜑p𝜆qq “ 0 (blue), the sign of Rep2𝜑p𝜆qq (gray), and the stationary contour Imp2𝜑q “ ˘1 (red). The deformation of RHP 20 is done in the regions determined by the stationary curves of 𝜑p𝜆q. More precisely, Let 𝜆 “ 𝜉 ` i𝜂, where 𝜉, 𝜂 P R. Then, Rep2𝜑p𝜆qq “ 𝜂 ` 1 ´ 𝜉 2 ´ 𝜂 2 ˘ 2 p𝜉 2 ` 𝜂 2q and Imp2𝜑p𝜆qq “ 𝜉 ` 1 ` 𝜉 2 ` 𝜂 2 ˘ 2 p𝜉 2 ` 𝜂 2q (6.1.2) . Clearly, 𝜑p𝜆q is purely imaginary o… view at source ↗
Figure 21
Figure 21. Figure 21: Definition of the matrix 𝑊˚p𝜆q. The solid contour is the stationary contour of 2𝜑p𝜆q, see [PITH_FULL_IMAGE:figures/full_fig_p068_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The contour Γ ˚ and the jump matrices 𝐺𝑊˚ p𝜆q for RHP 21. RHP 21(2) readily follows from the factorizations obtained above and RHP 20(2). The limit at 0 in RHP 21(3) is simply the result of the definition of 𝑊˚p𝜆q around the origin and of RHP 20(3). The limit at infinity in RHP 21(3) is the consequence of the fact that [PITH_FULL_IMAGE:figures/full_fig_p068_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Contour Γ𝑃𝑐, which consists of the coordinate axes and the ray argp𝑧q “ ´𝜋{4, and the jump matrices 𝐺𝑃𝑐p𝑧q for RHP 22. RHP 22 can be explicitly solved with the help of parabolic cylinder functions, see for example [IMY25, Section 4.2.3] (and recall that 𝐵 R “ 𝑒 ´𝜘𝑥). Its explicit form is also given further below in Appendix D. There, we explain that the asymptotics of 𝑃𝑐p𝑧q in RHP 22(3) can be improved to… view at source ↗
Figure 24
Figure 24. Figure 24: The contour Γ𝑉 for RHP 25. Following dressing technique of [BI12], we look for a solution of RHP 25 in the form 𝑉 ˚ p𝜆q “ p𝜆𝐼 ` 𝐵q𝑉p𝜆q ˆ 𝜆 ´ 1 0 0 𝜆 ` 1 ˙´1 (6.4.3) , where the matrix 𝐵 will be specified shortly and 𝑉p𝜆q solves the following (small norm) Riemann-Hilbert problem. Riemann-Hilbert Problem 26. Find a 2 ˆ 2 matrix function 𝑉p𝜆q such that (1) 𝑉p𝜆q is analytic in CzΓ𝑉 ; (2) one-sided traces 𝑉˘p𝜆… view at source ↗
read the original abstract

We consider solutions of the sinh-Gordon Painlev\'e III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty)$. They are parametrized by the monodromy parameter $p\in\overline{\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\mathbb{R}}$ when $p=\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\to\infty$. Here, we describe the transition as $x, p\to \infty$, $2\Im(p)=-s^{\mathbb R}$, between singular solutions ($|p|<\infty$) and smooth solutions ($p=\infty$). In short, if we parametrize $|p|^2 = 1 + e^{2\varkappa x}$, then the smooth exponential asymptotics of the solutions extends to the region $\varkappa>1$, with a change of the leading order term at $\varkappa=2$; at $\varkappa=1$ the exponential behavior transitions into an elliptic asymptotics, which holds for all $0<\varkappa<1$; as $\varkappa$ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all $p$ fixed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript examines real solutions on (0,∞) of the sinh-Gordon Painlevé III equation u_xx + (1/x)u_x = sinh u, parametrized by the monodromy datum p ∈ ℂ̄ with |p|>1 together with an auxiliary real parameter s^ℝ when p=∞. Building on prior joint work with Its that treated the x→∞ asymptotics for fixed p, the paper analyzes the joint limit x,p→∞ subject to 2 Im(p)=-s^ℝ. Under the auxiliary scaling |p|^2=1+e^{2ϰ x} it asserts four regimes: smooth exponential asymptotics persist for ϰ>1 (with a change of leading coefficient at ϰ=2), an elliptic regime occupies 0<ϰ<1, and the elliptic description degenerates to a trigonometric one as ϰ→0 (recovering the fixed-p case).

Significance. If the claimed transition asymptotics are rigorously established, the work supplies a continuous interpolation between the singular (|p|<∞) and smooth (p=∞) families that was missing from the earlier fixed-p analysis. Such a parametrization-dependent unification is potentially useful for applications in integrable systems and random-matrix theory that require control across the full range of monodromy data.

major comments (2)
  1. [Abstract] Abstract: the central scaling |p|^2=1+e^{2ϰ x} together with the reality link 2 Im(p)=-s^ℝ is asserted to produce a continuous transition, yet no derivation or matching argument is supplied showing how this choice arises from the underlying Riemann-Hilbert or isomonodromic problem; without that step the claim that the scaling preserves the reality condition on (0,∞) remains unverified.
  2. [Abstract] Abstract: the transition points ϰ=2 and ϰ=1 are stated to mark changes in leading-order behavior, but no error estimates, uniformity statements, or matching conditions between the exponential, elliptic, and trigonometric regimes are indicated; these are load-bearing for the asserted continuity of the transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. The comments correctly identify areas where the abstract could better motivate the scaling and clarify the technical underpinnings of the transitions. We respond point-by-point below and will incorporate clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central scaling |p|^2=1+e^{2ϰ x} together with the reality link 2 Im(p)=-s^ℝ is asserted to produce a continuous transition, yet no derivation or matching argument is supplied showing how this choice arises from the underlying Riemann-Hilbert or isomonodromic problem; without that step the claim that the scaling preserves the reality condition on (0,∞) remains unverified.

    Authors: We agree that the abstract does not derive the scaling. The parametrization |p|^2 = 1 + e^{2ϰ x} is chosen so that the deviation from the unit circle grows exponentially in x, thereby interpolating between the fixed-p (singular) regime recovered as ϰ → 0 and the p = ∞ (smooth) regime recovered as ϰ → ∞. The auxiliary condition 2 Im(p) = -s^ℝ is imposed directly on the monodromy data to guarantee that the associated solution u remains real-valued on (0,∞). A short derivation showing how this scaling emerges from the underlying Riemann-Hilbert problem (via the isomonodromic deformation equations) will be added to the introduction of the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: the transition points ϰ=2 and ϰ=1 are stated to mark changes in leading-order behavior, but no error estimates, uniformity statements, or matching conditions between the exponential, elliptic, and trigonometric regimes are indicated; these are load-bearing for the asserted continuity of the transition.

    Authors: The referee correctly observes that the abstract omits error estimates, uniformity statements, and explicit matching conditions. The body of the manuscript derives the leading-order asymptotics separately in each regime via steepest-descent analysis of the Riemann-Hilbert problem. To substantiate the claimed continuity, we will add a dedicated paragraph (or short subsection) that outlines the matching procedure across the critical values ϰ = 2 and ϰ = 1, together with the available error bounds and the regions of uniformity. This material will be placed immediately after the statement of the main results. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior asymptotics; independent scaling parameter introduced

full rationale

The paper cites its prior joint work with A. Its solely for the base large-x asymptotics of real solutions. The transition analysis is performed by adopting the explicit parametrization |p|^2 = 1 + e^{2ϰ x} together with the reality link 2 Im(p) = -s^R; this scaling is presented as a choice that captures the continuous transition between singular and smooth regimes and is not derived from or fitted to the target asymptotic quantities. No load-bearing step reduces by construction to a self-citation, a fitted input renamed as prediction, or a self-definitional relation. The new regimes (ϰ > 1, ϰ = 1 transition, 0 < ϰ < 1 elliptic, ϰ → 0 trigonometric) therefore retain independent content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis rests on the monodromy parametrization of real solutions and on the validity of the earlier large-x asymptotics; the scaling parameter ϰ is introduced to organize the transition but is not derived from first principles within the abstract.

free parameters (1)
  • ϰ
    Scaling parameter introduced via the relation |p|^2 = 1 + e^{2ϰ x} to interpolate between singular and smooth regimes.
axioms (1)
  • domain assumption Real solutions on (0, ∞) are parametrized by p ∈ ℂ-bar with |p| > 1 and, when p = ∞, by an additional real parameter s^R.
    Stated directly in the abstract as the starting point for the asymptotic analysis.

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