Rational Solutions of the Painlev\'e-III Equation: Large Parameter Asymptotics

The Painlev\'e-III equation with parameters $\Theta_0=n+m$ and $\Theta_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of $E$ but blows up near the origin, which is the only fixed singularity of the Painlev\'e-III equation. In both the interior and exterior domains we provide accurate asymptotic formul\ae\ for $u_n(x;m)$ that we compare with $u_n(x;m)$ itself for finite values of $n$ to illustrate their accuracy. We also consider the exceptional cases where $m$ is a half-integer, showing that the poles and zeros of $u_n(x;m)$ now accumulate along only one or the other of two"eyebrows", i.e., exterior boundary arcs of $E$.


INTRODUCTION
Generic solutions of the six Painlevé equations cannot be expressed in terms of elementary functions, hence the common terminology of Painlevé transcendents for the general solutions of these famous equations. However, it is also known that all of the Painlevé equations except for the Painlevé-I equation admit solutions expressible in terms of classical special functions (e.g., Airy solutions for Painlevé-II, or Bessel solutions for Painlevé-III) as well as rational solutions, both of which occur for certain isolated values of the auxiliary parameters (each Painlevé equation except Painlevé-I is actually a family of differential equations indexed by one or more complex parameters). Rational solutions of Painlevé equations have attracted interest partly because they are known to occur in several diverse applications such as the description of equilibrium configurations of fluid vortices [11] and of particular solutions of soliton equations [10], electrochemistry [1], parametrization of string theories [15], spectral theory of quasi-exactly solvable potentials [19], and the description of universal wave patterns [6]. In several of these applications it is interesting to consider the behavior of the rational Painlevé solutions when the parameters in the equation become large (possibly along with the independent variable); as the degree of the rational function is tied to the parameters via Bäcklund transformations, in this limit algebraic representations of rational solutions become unwieldy and hence less attractive than analytical ones as a means for extracting asymptotic behaviors. Recent progress on the analytical study of large-degree rational Painlevé solutions includes [3,7,8,17] for Painlevé-II and [5,16] for Painlevé-IV. Both of these equations have the property that there is no fixed singular point except the point at infinity. On the other hand, the Painlevé-III equation is the simplest of the Painlevé equations having a finite fixed singular point (at the origin). This paper is the second in a series beginning with [4] concerning the large-degree asymptotic behavior of rational solutions to the Painlevé-III equation, which we take in the generic form It is convenient to represent the constant parameters Θ 0 and Θ ∞ in the form Θ 0 = + and Θ ∞ = − + 1.
It is known that if ∈ ℤ, there exists a unique rational solution ( ) = ( ; ) of (1) that tends to 1 as → ∞. The odd reflection ( ) = − (− ; ) provides a second distinct rational solution. Similarly, if ∈ ℤ, there are two rational solutions tending to ±i as → ∞, namely ( ) = ±i (±i ; ), while if neither nor is an integer, (1) has no rational solutions at all. If only one of and is an integer, then there are exactly two rational solutions; however if both ∈ ℤ and ∈ ℤ there are exactly four distinct rational solutions: ( ; ), − (− ; ), i (i ; ), and −i (−i ; ).

Algebraic representation.
It has been shown [9,12,20] that ( ; ) admits the representation , For not too large, the recurrence relation (4) provides an effective computational strategy to obtain the poles and zeros of ( ; ). The rational function ( ; ) has the following symmetry: . (5) This follows from the fact that ( ) ↦ (− ) −1 is a symmetry of (1)- (2) corresponding to the parameter mapping ( , ) ↦ (− , ). Since this symmetry preserves rationality and asymptotics → 1 as → ∞, it descends from general solutions to the particular solution ( ; ) as written in (5). 1.1.2. Analytic representation. The goal of this paper is to study ( ; ) when is a large positive integer and is a fixed complex number. The representation (3) is useful to determine numerous properties of the rational Painlevé-III solutions, however when is large another representation becomes more preferable. To explain this alternate representation, we first define some -dependent arcs in an auxiliary complex -plane as follows. Given ∈ ℂ with ≠ 0 and |Arg( )| < , there is an intersection point and four oriented arcs ∞ ⬔ , 0 ⬔ , ∞ ⬕ , and 0 ⬕ such that: • The arc ∞ ⬔ originates from = ∞ in such a direction that i is negative real and terminates at = , the arc 0 ⬔ begins at = and terminates at = 0 in a direction such that −i −1 is negative real, and the net increment of the argument of along ∞ ⬔ ∪ 0 ⬔ is Δ arg(⬔) = 2Arg( ) ± 2 .
The ambiguity of the sign in (6) will be explained below (see Remark 1). • The arc ∞ ⬕ originates from = ∞ in such a direction that −i is negative real and terminates at = , the arc 0 ⬕ begins at = and terminates at = 0 in a direction such that i −1 is negative real, and the net increment of the argument of along ∞ ⬕ ∪ 0 ⬕ is Δ arg(⬕) = 2Arg( ).
• The arcs ∞ ⬔ , 0 ⬔ , ∞ ⬕ , and 0 ⬕ do not otherwise intersect. See Figure 19 below for an illustration in the case of Arg( ) = 1 4 . We define a single-valued branch of the argument function ↦ arg( ), henceforth denoted arg ⬕ ( ), by first selecting ∞ ⬕ ∪ 0 ⬕ as the branch cut, and then defining arg ⬕ ( ) = 0 for sufficiently large positive when Im( ) > 0 and arg ⬕ ( ) = for sufficiently large negative when Im( ) < 0. It is easy to see that this definition is consistent for > 0 but there is a jump across the negative real -axis. We define an associated branch of the complex logarithm log( ) by setting log ⬕ ( ) ∶= ln | | + i arg ⬕ ( ). Then, given ∈ ℂ, the corresponding branch of the power function will be denoted by ⬕ ∶= e log ⬕ ( ) . Finally, we denote by the union of the four oriented arcs ∞ ⬔ , 0 ⬔ , ∞ ⬕ , and 0 ⬕ , and define the function The following Riemann-Hilbert problem was formulated in [4,Sec. 1.2]. Here and below we follow the convention that subscripts +/− refer to boundary values taken on a jump contour from the left/right, and 3 ∶= diag[1, −1] denotes a standard Pauli spin matrix.

Analyticity: ↦ ( ) is analytic in the domain ∈ ℂ ⧵ . It takes continuous boundary values on ⧵ {0}
from each maximal domain of analyticity.

Jump conditions:
The boundary values ± ( ) are related on each arc of by the following formulae: 3. Asymptotics: ( ) → as → ∞. Also, the matrix function ( ) ⬕ has a well-defined limit as → 0 (the same limit from each side of ). (6), the sign may be reversed by a surgery performed on ∞ ⬔ ∪ 0 ⬔ for any given value of ≠ 0, |Arg( )| < which leaves the conditions of Riemann-Hilbert Problem 1 invariant. The surgery consists of bringing ∞ ⬔ together (with the same orientation) with 0 ⬔ in some small arc. The jump for cancels on this small arc because the jump matrices in (9)- (10) are inverses of each other; thus, up to some relabeling, one has effectively changed the sign in (6). In [4] the choice of sign in (6) was tied to the sign of Im( ) due to the derivation of Riemann-Hilbert Problem 1 from direct/inverse monodromy theory, however the above surgery argument shows that the sign is in fact arbitrary. The freedom to choose this sign will be important later when the solution of Riemann-Hilbert Problem 1 is constructed for large .

Remark 1. Given any choice of sign in
It turns out that if Riemann-Hilbert Problem 1 is solvable for some ∈ ℂ ⧵ {0}, then we may define corresponding matrices ∞ 1 ( ) and Then, according to [4,Theorem 1], an alternate formula for the rational solution ( ; ) of the Painlevé-III equation where we have suppressed the parametric dependence on ∈ ℤ and ∈ ℂ on the right-hand side.

Results and outline of paper.
A good way to introduce our results is to first explain a simple formal asymptotic calculation. Since we are interested in solutions = ( ; ) of (1) with parameters written in the form (2) when is large, and since numerical experiments such as those in [4,Sec. 2] suggest that the largest poles and zeros of ( ; ) lie at a distance | | from the origin proportional to with a local spacing that neither grows nor shrinks with , it is natural to introduce a complex parameter ≠ 0 and a new independent variable ∈ ℂ by setting = + . It follows that if ( ) solves (1)-(2), then ( ) ∶= −i ( = + ) satisfies in which the ( −1 ) symbol absorbs several terms each of which is explicitly proportional to −1 ≪ 1. Dropping these formally small terms leads to an autonomous second-order equation which is amenable to classical analysis: wherė denotes a formal approximation to . Solutions of the equation 1 (17) can be classified as follows: • Equilibrium solutionṡ ≡ constant. Generically with respect to there are four such equilibria:̇ ≡ ±1 anḋ where to be precise we take the square roots to be equal to 1 at = ∞ and to be analytic in except on a line segment branch cut connecting the branch points = ± 1 2 i in the parameter plane. Note that of these four, the unique equilibrium that tends to −i as → ∞ (as would be consistent with = ( ; ) → 1 as → ∞) iṡ ≡ + 0 ( ). • Non-equilibrium solutions. These can be obtained by integrating (17) to find a first integral. Thus, provideḋ ( ) is non-constant, we may write (17) in the equivalent form in which is a constant of integration. There are two types of non-equilibrium solutions: -If is generic given such that has 4 distinct roots, then all non-constant solutions of (19) are (doublyperiodic) elliptic functions of with elliptic modulus depending on and . -If = ( ) is such that the quartic has fewer than 4 distinct roots, then the higher-order roots are necessarily equilibrium solutions of (17) and all non-constant solutions of (19) are (singly-periodic) trigonometric functions of .
Our rigorous analysis of ( ; ) in the large-limit shows that all of the above types of solutions of the approximating equation (17) play a role. In order to begin to explain our results, first observe that if is replaced with + , then for large , the dominant factors in the off-diagonal elements of the jump matrices in Riemann-Hilbert Problem 1 are the exponentials e ± ( ; ) , where ( ; ) ∶= − log( ) − i ( ).
The fact that is multi-valued is not important because e ± ( ; ) is single-valued whenever ∈ ℤ. However, Re( ( ; )) is certainly single-valued for ∈ ℂ ⧵ {0} and ∈ ℂ. For simplicity, in the rest of the paper we write ( ) ∶= + 0 ( ). Since ( ) is analytic and non-vanishing in its domain of definition, the left-hand side of the equation Re( ( ( ); )) = 0 (21) defines a harmonic function in the complex -plane omitting the vertical branch cut of ( ) connecting the branch points ± 1 2 i. Therefore, (21) determines a curve in the latter domain that turns out to be the union of four analytic arcs: two rays on the imaginary axis connecting the branch points = ± 1 2 i to = ±i∞ respectively, an arc in the right half-plane joining the two branch points, and its image under reflection through the imaginary axis. The union of the latter two arcs is the boundary of a compact and simply-connected eye-shaped set denoted containing the origin = 0. The eye is symmetric with respect to reflection through the origin as well as both the real and imaginary axes. See Figure 20 below. Our first result is then the following.
Thus, is approximated by the unique equilibrium solution of (17) that tends to −i as → ∞, provided that lies outside the eye . Since ( ) is analytic and non-vanishing as a function of bounded away from , the uniform convergence immediately implies the following. Corollary 1. Fix ∈ ℂ and let be as in the statement of Theorem 1. Then (⋅; ) has no zeros or poles in the set for sufficiently large. 1 More properly, it is a family of equations parametrized by ∈ ℂ ⧵ {0}.
As an application of these results, let ∈ ℂ ⧵ and let denote a positively-oriented loop surrounding the point . Then, from Cauchy's integral formula it follows that, as → +∞, where to evaluate the integral we used (22). It is easy to see that the error term enjoys similar uniformity properties as in Theorem 1.
Theorem 2 (Elliptic asymptotics of ( ; )). Fix ∈ ℂ ⧵ (ℤ + 1 2 ). For each ∈ ℤ ≥0 and each ∈ R , the functioṅ ( ) ∶= −i̇ ( , ; ) is a non-equilibrium elliptic function solution of (17) in the form (19) with integration constant = ( ). If > 0 is an arbitrarily small fixed number and ⊂ R and ⊂ ℂ are compact sets, then holds uniformly on the set of ( , , ) defined by the conditions ∈ , ∈ such that Under the same conditions and with the same sense of convergence, which provides asymptotics of ( ; ) when ∈ L .
The formula (28) follows from (26) with the use of the symmetry (5) (and thaṫ ( , ; ) is bounded and bounded away from zero on the indicated set, as it happens). Thus, provided that −1 lies in either domain L or R and is not a half-integer, the rational Painlevé-III function ( ; ) is locally approximated by a non-equilibrium elliptic function solution of the differential equation (17). Note that the fact that the leading term on the right-hand side of (28) is an elliptic function follows from the first statement of Theorem 2 and the fact that the integrated form (19)  it is the scale on which ( ; ) resembles a fixed elliptic function. On the other hand the variable captures the way that the elliptic modulus depends on the point of observation within the eye and unlike the meromorphic dependence on ,̇ ( , ; ) is a decidedly non-analytic function of . If we approximate ( ; ) by setting = 0 and letting vary, we obtain a globally accurate (on ) approximation that is unfortunately not analytic in . However if we fix ∈ R and let vary, we obtain a locally accurate ( ∈ , so − = = (1) as → +∞) approximation that is an exact elliptic function depending only parametrically on .
If in any of the conditions (27) we put = 0, then the corresponding phase agrees with a point of the lattice 2 iℤ + ( )ℤ and the associated factor in the definition oḟ ( , ; ) vanishes. For > 0, each condition in (27) defines a "swiss-cheese"-like region in the variables ( , ) given ∈ ℤ ≥0 and ∈ ℂ ⧵ (ℤ + 1 2 ) with holes centered at points corresponding to lattice points. In fact, if ∈ R is also fixed, then the lattice 2 iℤ + ( )ℤ is a uniform lattice and each of the conditions in (27) omits from the complex -plane the union of disks of radius centered at the lattice points. On the other hand, if instead it is ∈ ℂ that is fixed, then each of the conditions (27) omits from the complex -plane neighborhoods of diameter proportional to −1 containing the points in a set that can be roughly characterized as a curvilinear grid of spacing proportional to −1 .
is a sequence such that  • ( , 0; ) = 0 for = , + 1, … (or such that  • ( , 0; ) = 0 for = , + 1, … ), then for each sufficiently small > 0 there is exactly one simple zero, and possibly a group of an equal number of additional zeros and poles, of ( ; ) is a sequence such that  • ( , 0; ) = 0 for = , + 1, … (or such that  • ( , 0; ) = 0 for = , + 1, … ), then for each sufficiently small > 0 there is exactly one simple pole, and possibly a group of an equal number of additional zeros and poles, of ( ; ) within The proof of this result depends on Theorem 2 and some additional technical properties of the zeros of the factors in the formula (25) and will be given in Section 4.7. The proof is based on an index argument, which computes the net number of zeros over poles within a small disk. For this reason, we cannot rule out the possible attraction of one or more pole-zero pairs of the rational function ( ; ), in excess of a simple zero (or pole), toward a given zero (or singularity) of the approximating function. However, we do not observe any such "excess pairing" in practice. One approach to ruling out any excess pairing would be to compare against precise counts of the zeros and poles of ( ; ) as documented in [12]. However, such a comparison would require accurate approximations in domains that completely cover the eye without overlaps. In this paper we avoid analyzing ( ; ) near the origin, the corners = ± 1 2 i, and the "eyebrows" (except in the special case ∈ ℤ + 1 2 ; see below). These are projects for the future. Although for these reasons there remains some ambiguity about the distribution of poles and zeros of the rational function ( ; ), our analysis gives very detailed information about the distribution of singularities and zeros of the approximatioṅ ( , ; ). In particular, we have the following.
There is a continuous function ∶ R → ℝ + , ∈ 1 loc ( R ), such that for any compact set ⊂ R , where d ( ) denotes Lebesgue measure in the -plane. The density is independent of ∈ ℂ ⧵ (ℤ + 1 2 ) and satisfies ( ) → 0 as → R ⧵ {0} and ( e i ) = ℎ( ) −1 + ( −1 ) as ↓ 0 for some function ℎ ∶ (− ∕2, ∕2) → ℝ + . We would expect that the same statement holds witḣ ( , 0; ) replaced by ( ; ), but this would require ruling out the excess pairing phenomenon mentioned above. The density function ( ) is defined in (211) below, and the proof of Theorem 3 is given in Section 4.7. Although the proof of Theorem 3 does not allow us to consider sets that depend on in any serious way, the assumtion that (29) holds when is the disk of radius −2 centered at the origin leads to the prediction that this disk contains (1) zeros/singularities oḟ ( , 0; ) consistent with the empirical observation that the smallest zeros and poles of ( ; ) scale like −1 in the -plane [4].
While the asymptotic approximations of the rational Painlevé-III function ( ; ) for −1 ∈ L ∪ R are much more complicated than the simple formula i ( −1 ) valid for −1 ∈ ℂ ⧵ , they are easily implemented numerically, once the necessary ingredients developed as part of the proof of Theorem 2 are incorporated. To quantitatively illustrate the accuracy of the approximations described in Theorems 1 and 2, we compare ( ; ) with its approximations for restricted to a real interval that bisects in Figures 1-3. In these figures, we found it compelling to plot the Comparison of ( ; ) (blue curve) with its approximations over the interval −0.5 < < 0.5 for = 0 with = 10 (left) and = 20 (right). The points where this interval intersects are shown with vertical gray lines. The approximatioṅ ( , 0; ) of Theorem 2 is plotted in between the gray lines with black broken curves. The dotted curve is the analytic continuation into from the right of the outer approximation i ( ) described in Theorem 1. Likewise, the dash/dotted curve is the meromorphic continuation into from the left of the same outer approximation. approximate formula i ( ) of Theorem 1 continued into the eye from the left and right, even though we have no basis for comparing the graphs of these (reciprocal) continuations with that of ( ; ) when ∈ . Indeed, in some situations these graphs appear to form quite accurate upper or lower envelopes of the wild modulated elliptic oscillations of ( ; ) that occur when ∈ and that are captured with locally uniform accuracy bẏ ( , 0; ). We have no explanation for these somewhat imprecise observations, but we find them interesting and note that similar phenomena occur for the rational solutions of the Painlevé-II equation (also without explanation) as was noted in [7]. Now, we go into the complex -plane where we can illustrate both the shape of the eye and the phenomenon of attraction of poles and zeros of ( ; ) to the left ( L ) and right ( R ) halves. In these figures, the zeros and poles of the rational Painlevé function ( ; ) are plotted with the following convention (as in our earlier paper [4]): • Zeros of ( ; ) that are also zeros of ( ; − 1): blue filled dots.
• Poles of ( ; ) that are also zeros of ( ; ): red filled dots.  1 5 i. Here the top row compares the real parts and the bottom row compares the imaginary parts (the graph of Im( ( ; )) is shown with a brown curve).
In addition to displaying the overall attraction of the poles and zeros to the eye domain , the plots in  are also intended to demonstrate the remarkable accuracy of the approximation of Theorem 2 in capturing the locations of individual poles and zeros as described in Corollary 2. As described in Section 4.7 below, each of the four factors in the fraction on the right-hand side of (25) has zeros that may be characterized as the intersection points of integral level curves of two different functions (see (202) and (203) below) defined on R (and via the symmetry (5), L ). We plot the families of level curves for each of the four factors in separate figures in order to demonstrate another phenomenon that is evident but for which we have no good explanation: the zeros of the separate factors in the approximatioṅ ( , 0; ) as defined by (25) appear to correspond precisely to the actual zeros of the four polynomial factors in the formula (3) for the rational Painlevé-III function ( ; ). This coincidence is what motivates the superscript notation (• versus •) on the four factors in (25); the zeros of the factors with superscript • (resp., •) apparently correspond in the limit → +∞ to filled (resp., unfilled) dots. Another feature of the plots in Figures 4-15 is that only one pole or zero is evidently attracted to each crossing point of the curves, which suggests that the excess pairing phenomenon that cannot be ruled out by our index-based proof of Corollary 2 does in fact not occur. Finally, these plots illustrate the most important properties of the pole/zero density function ( ) described in Theorem 3, namely the infinite density at the origin and the dilution of poles/zeros near the boundaries of L and R (which include the imaginary axis vertically bisecting ).
Clearly, when ∈ ℂ ⧵ (ℤ + 1 2 ) there are many poles and zeros in the domains L and R when is large, and in this situation we say that the eye is open. On the other hand, the large-asymptotic behavior of ( ; ) when −1 is in a neighborhood of the eye is completely different than described above when ∈ ℤ + 1 2 . We refer to the closures (i.e., including endpoints) of the arcs of L and R in the open left and right half-planes respectively as the "eyebrows"  of the eye , denoting them by 0 ⬕ and ∞ ⬔ , respectively. Our first result is that, in a sense, the eye is closed when ∈ ℤ + 1 2 .
Theorem 4 (Equilibrium asymptotics of ( ; ) for ∈ ℤ + 1 2 ). Suppose that = −( where ∞ ⬔ ( ) denotes the meromorphic continuation of ( ) from a neighborhood of = ∞ to the maximal domain ℂ ⧵ ∞ ⬔ as a non-vanishing function whose only singularity is a simple pole at the origin = 0, and the error estimate is uniform for ∈ . Likewise, if = + 1 2 for ∈ ℤ ≥0 and ⊂ ℂ ⧵ 0 ⬕ is bounded away from 0 ⬕ , then   where 0 ⬕ ( ) denotes the analytic continuation of ( ) from a neighborhood of = ∞ to the maximal domain ℂ ⧵ 0 ⬕ as a function whose only zero is simple and lies at the origin, and the error estimate is uniform for ∈ . The functions ∞ ⬔ ( ) and 0 ⬕ ( ) both agree with ( ) for ∈ ℂ ⧵ , and they are reciprocals of one another when ∈ . Theorem 4 is proved in Section 5.1. Note that this result is consistent with Theorem 1, which does not require any condition on ∈ ℂ. Moreover, it gives a far-reaching generalization of Theorem 1 for the special case of ∈ ℤ+ 1 2 . The uniform nature of the convergence implies that ( ; ) can have no poles or zeros in for sufficiently large , unless the set contains the origin, in which case an index argument predicts a unique simple pole near the origin for = −( + 1 2 ) and a unique simple zero near the origin for = + 1 2 . However, it is proven in [12] that there is a simple pole or zero exactly at the origin if is sufficiently large (given ∈ ℤ ≥0 ). Therefore, we have the following.  This result can be combined with Theorem 4 to show immediately as in (23) that the convergence of ( ; ) for ∈ extends to all derivatives. Corollary 3 also shows that if ∈ ℤ + 1 2 , all of the poles/zeros but one are attracted toward one or the other of the eyebrows as → +∞, depending on the sign of ; this is what we mean when we say that the eye is closed. Counting arguments suggest it is reasonable that the poles and zeros should be organized near curves rather than in a two-dimensional area such as L ∪ R in this case. Indeed, in [12] it is also shown that the total number of zeros and poles of ( ; ) scales as as → +∞ when ∈ ℤ + 1 2 , while for ∈ ℂ ⧵ (ℤ + 1 2 ) the number scales as 2 . Our methods allow for the following precise statement concerning the nature of convergence of the poles/zeros to one or the other of the eyebrows for ∈ ℤ + 1 2 . The following results refer to a "tubular neighborhood" of the eyebrow ∞ ⬔ defined as follows: for sufficiently small positive constants 1 and 2 , Since points on the eyebrow Theorem 5 (Layered trigonometric asymptotics of ( ; ) for ∈ ℤ + 1 2 ). Let = −( 1 2 + ), ∈ ℤ ≥0 , and let be as defined in (32). Then the following asymptotic formulae hold in which the error terms are uniform on the indicated sub-domains of from which small discs of radius proportional to an arbitrarily small multiple of −1 centered at each zero or pole of the indicated approximation are excised: • If ∈ with Re( ( ( ) −1 ; )) ≤ − 1 2 −1 ln( ), then ( ; ) =̇ + ( −1 ) wherė is given explicitly by (282). These results imply corresponding asymptotic formulae for ( ; ) if = 1 2 + , ∈ ℤ ≥0 by the exact symmetry (5); in particular the eyebrow near which the asymptotics are nontrivial is then the left one, 0 ⬕ . The inequalities on in the statement of the theorem describe a dissection of into finitely-many (depending on ) "layers" roughly parallel to the right eyebrow ∞ ⬔ and overlapping at their common boundaries. The order of the layers as written in the theorem corresponds to crossing ∞ ⬔ from inside to outside, and the "interior" layers described by the index are each of width proportional to −1 ln( ). The approximatioṅ assigned to each layer is a fractional linear (Möbius) function of e 2 ( ( ) −1 ; ) where the power and the coefficients of the linear expressions in the numerator/denominator depend on the layer. The latter coefficients are relatively slowly-varying functions of alone that are explicitly built from ( ), and hence the dominant local behavior in any given layer is essentially trigonometric with respect to . We wish to stress that, unlike the approximation formula (25) whose ingredients involve implicitlydefined functions of ∈ R and elements of algebraic geometry, the approximatioṅ in each layer is an elementary function of ( ; ) and ( ). In particular, it is easy to check that when is in the innermost or outermost layers but bounded away from ∞ ⬔ (the "overlap domain"), Theorem 5 is consistent with Theorem 4.
The analogue of Corollary 2 in the present context is the following. As before, we suspect that with additional work one should be able to preclude the excess pairing phenomenon, so that the poles and zeros of ( ; ) and its approximatioṅ are in one-to-one correspondence. Now in each layer of , the poles and zeros oḟ are easily seen to lie exactly along certain explicit curves roughly parallel to the eyebrow. Analogous results hold for the approximation to ( ; ) for = 1 2 + , ∈ ℤ ≥0 , obtained froṁ via the symmetry Corollary 4 and Theorem 6 are proved in Section 5.2.8. To illustrate the accuracy of these results, we compare the exact locations of zeros and poles of ( ; ) for = −( + 1 2 ), ∈ ℤ ≥0 , with the curves described in Theorem 6 in Figures 16-18. In addition to illustrating the accuracy of the approximation bẏ , these figures demonstrate another phenomenon for which we do not yet have an explanation: for any given curve, the poles/zeros attracted are those contributed by exactly one of the four polynomial factors in (3). Furthermore, there appears again to be no excess pairing of poles and zeros.
Evidently, the large-asymptotic behavior of ( ; ) is completely different for = ±( + 1 2 ), ∈ ℤ ≥0 , and for = ±( + 1 2 ) + , however small ≠ 0 is. In other words, even crude aspects of the large-asymptotic behavior of ( ; ) for −1 in a neighborhood of the eye fail to be uniformly valid with respect to the second parameter near half-integer values of the latter. Thus, given ∈ ℂ, the eye is either open or closed in the large-limit. On the other hand, the polynomials ( ; ) in the formula (3) are actually polynomials in both arguments and [12], and in this sense the limits of → +∞ and → ℤ + 1 2 do not commute. Capturing the process of the closing of the eye requires connecting with in a suitable double-scaling limit so that tends to a given half-integer as → +∞. In a subsequent paper, we will show that in the right double-scaling limit, all three types of solutions of the autonomous model equation (17) play a role in describing ( ; ) as → +∞.

SPECTRAL CURVE AND -FUNCTION
When is large, the exponential factors e ± ( ; ) appearing in the jump conditions (9)-(12) need to be balanced in general by some compensating factors that can be used to control exponential growth. We therefore introduce a " -function" ( ; ) that is taken to be bounded and analytic in ℂ ⧵ with ( ; ) → ∞ ( ) as → ∞ for some ∞ ( ) to be determined, and we set we place the following conditions on . We want to be chosen so that can be deformed and then split into several arcs along each of which one of the following alternatives holds (recall that is defined by (20)): and hence if ≠ 0, Therefore, if = 0, Liouville's theorem shows that while if ≠ 0 we necessarily have that where (⋅; , ) is the quartic polynomial defined by (19) and it only remains to determine . Since the zero locus of ( ; , ) is obviously symmetric with respect to the involution ↦ −1 , the following configurations for ( ; , ) include all possibilities, given that ≠ 0: Comparing with (19) shows that this is possible for all ≠ 0, provided that is determined as a function of up to reciprocation by + −1 = i −1 and then is given the value = − 1 4 2 ( 2 +4+ −2 ) = − 1 4 (2 2 −1). In this case, ( ; , ) is again a perfect square and hence either ′ ( ; ) − 1 2 Only the former is consistent with (20) given that ′ ( ; ) = ( −2 ) as → ∞ and again we deduce that ′ ( ; ) = 0 and hence also ( ; ) = ∞ ( ). This turns out to be the case corresponding to ∈ ℂ ⧵ . (iii) There is one double root and two simple roots, with the double root being fixed by the involution and hence occurring at = ±1 and the simple roots being permuted by the involution and hence being given by = 0 and = −1 Comparing with (19) shows that this configuration is possible for all ≠ 0, provided that 0 is determined up to reciprocation by 0 + −1 0 = 2i −1 ∓ 2 and that is assigned the value = − 1 2 2 (1 ± ( 0 + −1 0 )) = 1 2 2 ∓ i . This case turns out to be relevant only when There are four simple roots, none of which equal 3 1 or −1, in which case for some 0 and 1 with 2 0 ≠ 1, Comparing with (19) shows that this case is possible for all ≠ 0 with arbitrary , and that then 0 and 1 are determined up to reciprocation and exchange by the identities 0 + −1 0 This turns out to be the case for ∈ L ∪ R .

Boutroux integral conditions.
In order to ensure that the constant associated with each distinguished arc of is real, it is necessary in the above case (iv) to impose further conditions. Given and such that this is the case, let Γ = {( , ) ∶ 2 = −4 ( ; , )} be the genus-1 Riemann surface or algebraic variety associated with the equation 2 = −4 ( ; , ) in ℂ 2 with coordinates ( , ). Let ( , ) be a canonical homology basis on Γ and take concrete representatives that do not pass through the preimages on Γ of each of the two points = 0 or = ∞. Then we impose the Boutroux conditions where = + i , i.e., ∶= Re( ) and ∶= Im( ). It follows from (35) that the differential d has real residues at the two points of Γ over = 0 and the two points over = ∞; therefore taken together the conditions (39) do not depend on the choice of homology basis. We expect that the two real conditions (39) should determine and as functions of ∈ ℂ. Differentiation of the algebraic identity relating and gives Comparing with (19) shows that this situation cannot occur for ≠ 0. Comparing with (19) shows that this case is not possible from which it follows (since the paths and may be locally taken to be independent of and ) that (41) Therefore, the Jacobian determinant of the equations (39) equals Noting that −1 −2 d is a nonzero differential spanning the (one-dimensional) vector space of holomorphic differentials on Γ, it follows from [13, Chapter II, Corollary 1] that the above Jacobian is strictly negative under the assumption that the four roots of ( ; , ) are distinct. Thus, an application of the implicit function theorem allows us to extend any solution of the integral conditions (39) for which ( ; 0 , 0 + i 0 ) has distinct roots to a neighborhood of 0 on which and are smooth real-valued functions of satisfying ( 0 ) = 0 and ( 0 ) = 0 . In fact, one can show that the Jacobian determinant (42) blows up as the spectral curve degenerates, and it is in this way that the implicit function theorem ultimately fails.

ASYMPTOTICS OF ( ; ) FOR ∈ ℂ ⧵
In this section, we study Riemann-Hilbert Problem 1 with = (i.e., we set = 0) and assume that lies in a neighborhood of = ∞ to be determined.

Placement of arcs of and determination of
. We first show that for sufficiently large in magnitude, we may take = − 1 4 (2 2 − 1) and hence ( ; , ) has two double roots; therefore the spectral curve is reducible leading to ( ; ) = ∞ ( ) for a suitable value of the latter constant. For large we take the double root = ( ) where the (1) error term applies in the limit → ∞ uniformly for in compact subsets of ℂ ⧵ {0}. Taking into account that 2 ( ; ) − ( ; ) has a double zero at = ( ), one can show that if | | is sufficiently large, taking the common intersection point of all four contour arcs to be the point ( ), it is possible to arrange the arcs so that Re(2 ( ; ) − ( ; )) < 0 (resp., Re(2 ( ; ) − ( ; )) > 0) holds on ∞ ⬔ ∪ 0 ⬔ (resp., on ∞ ⬕ ∪ 0 ⬕ ), with the inequality being strict except at the intersection point = ( ), compare Figure 19. The function ( ) has an analytic continuation from the neighborhood of = ∞ to the maximal domain ∈ ℂ ⧵ , where denotes the imaginary segment connecting the two branch points ± 1 2 i. As is brought in from the point at infinity, it remains possible to place the arcs of the contour as described above at least until either meets the branch cut of ( ) or the topology of the zero level set of Re(2 ( ; )− ( ; )) changes. The latter occurs precisely when the only other critical point = ( ) −1 moves onto the zero level set; since Re( ( −1 ; )) = −Re( ( ; )), whenever both ( ) and ( ) −1 lie on the same level of Re( ( ; )) we necessarily have Re( ( ( ); )) = 0. The set of ∈ ℂ ⧵ where the latter condition holds true is plotted in Figure 20. Because ∈ iℝ ⧵ implies that | ( )| = 1, it is easy to confirm that indeed Re( ( ( ); )) = 0 for such , see also Figure 20. The rest of the points comprise a closed curve with two smooth arcs meeting at the branch points ± 1 2 i and bounding the eye-shaped domain defined in Section 1.2. The following figures illustrate how the domains such as shown in Figure 19 change as the value of varies near the arcs of the curve shown in Figure 20. Figure 21 concerns the three points on the real axis and Figure 22 concerns the three points on the diagonal. Figure 23 shows that although there is a topological change in the level curve as crosses the imaginary axis in the exterior of , this does not obstruct the placement of the contour arcs of . On the other hand, the topological change that occurs when lies along the arc of in the right half-plane (resp., left half-plane) only obstructs placement of the arc ∞ ⬔ (resp., 0 ⬕ ) and therefore we write as the union of two closed arcs: = ∞ ⬔ ∪ 0 ⬕ . Note that the surgery allowing for a sign change Δ arg(⬔) = 2Arg( ) − 2 ↔ Δ arg(⬔) = 2Arg( ) + 2 (see Remark 1) is compatible with the sign-chart/contour placement scheme provided that the domain  (6)) are also shown. is defined as the bounded region between the two black curves, bisected by the branch cut of ( ). The sign of Re( ( ( ); )) is indicated in each region. In particular, R (resp., L ) is the bounded region where Re( ( ( ); )) > 0 (resp., Re( ( ( ); )) < 0) holds.
Re(2 ( ; ) − ( ; )) < 0 consists of a single component. If it consists of two components, then the contours 0 ⬔ and ∞ ⬔ necessarily lie in distinct components and the surgery becomes impossible. The former holds in the exterior of for Re( ) > 0 and the latter for Re( ) ≤ 0.

Parametrix construction.
Let be fixed outside of , let > 0 be a fixed sufficiently small (given ) constant, and let denote the simply-connected neighborhood of = ( ) defined by the inequality |2 ( ; ) − ( ; )| < 2 . We will define a parametriẋ ( ; , ) for ( ; , ) in (33) by a piecewise formula: Noting that the jump matrix for ( ; , ) converges uniformly on ⧵ (with exponential accuracy) to except on 0 ⬕ , where the limit is instead −e 2 i 3 , and that ( ; , ) should have a limit as → 0, we define  Figure 20. The top row shows a neighborhood of the unit disk in the -plane, while the bottom row shows the exterior of the unit disk in the −1 -plane. In the plots in the right-hand column, the level curve has broken and it is no longer possible to place the contour arc ∞ ⬔ connecting ( ) and ∞ completely within the red region. This phase transition, which apparently occurs only on the right edge of the domain , is only relevant if the jump matrix on ∞ ⬔ is not the identity, i.e., if − 1 2 ∉ ℤ ≥0 . These plots show contours with Δ arg(⬔) = 2Arg( ) − 2 = −2 . The other choice Δ arg(⬔) = 2Arg( ) + 2 = 2 would also be compatible with the sign chart. out ( ; , ) as the following diagonal matrix: where the branch cut is taken to be 0 ⬕ and the branch is chosen such that the right-hand side tends to as → ∞. In order to definė in ( ; , ), we will find a certain canonical matrix function that satisfies exactly the jump conditions of ( ; , ) within the neighborhood and then we will multiply the result on the left by a matrix holomorphic in to arrange a good match witḣ out ( ; , ) on . For the first part, we introduce a conformal mapping ∶ → ℂ by the following relation: Im( ( ; )) < 0, Re( ( ; )) ≠ 0.   Recalling the precise definition of , with its cut along ∞ ⬕ ∪ 0 ⬕ , it follows that ( ; , ) can be continued to the whole domain as an analytic nonvanishing function. The jump conditions satisfied by ( ; , ) within are then the following: and Although we will only use its values for ∈ ℂ ⧵ , the outer parametriẋ out ( ; , ) has a convenient representation also for ∈ in terms of the conformal coordinate = ( ): where the power function of refers to the principal branch cut for < 0, and where ( ; , ) is holomorphic and nonvanishing in . Now letting ∶= 1∕2 ( ; ), we define precisely a matrix ( ; ) as the solution of the following model Riemann-Hilbert problem.

Riemann-Hilbert Problem 2. Given any
∈ ℂ, seek a 2 × 2 matrix function ↦ ( ; ) with the following properties: and Here, refers to the principal branch.
This problem will be solved in all details in Appendix A. From it, we define the inner parametriẋ in ( ; , ) as follows:̇ in ( ; , ) ∶= ( ; , ) 3 As shown in Appendix A, ( ; ) ( + 1 2 ) 3 has a complete asymptotic expansion in descending powers of as → ∞ (see (321)). Taking into account the explicit leading terms from the expansion (321) and using the fact that ( ; ) is bounded away from zero for ∈ , we geṫ . This matrix satisfies a jump condition of the form + ( ; , ) = − ( ; , )( + exponentially small) as → +∞ uniformly for ∈ (in fact, . To see this for ∈ ⧵ , note that where is the corresponding jump matrix foṙ out ( ; , ), which is just the diagonal part of ( ; , ) and which hence reduces to the identity matrix except on 0 ⬕ . The desired result then follows because ( ; , )̇ ( ; , ) −1 − is exponentially small in the limit → +∞, whilė out − ( ; , ) and its inverse are independent of and bounded because is excluded from . Finally, for ∈ , taken with clockwise orientation, we have The jump contour for ( ; , ) (and also for the related matrix ( ; , ) defined below) in a typical case of ∈ ℂ⧵ is shown in Figure 25. Taking into account the leading term on the upper off-diagonal in (59), we define a parametrix for ( ; , ) as a triangular matrix independent of : Since (⋅; , ) and (⋅; , ) are analytic and nonvanishing within , and since (⋅; ) is univalent on with ( ( ); ) = 0, the above Cauchy integral can be evaluated by residues. In particular, if ∈ ℂ ⧵ , theṅ Note thaṫ ( ; , ) is analytic for ∈ ℂ ⧵ ,̇ ( ; , ) → as → ∞, and across satisfies (by the Plemelj formula from (63)) the jump conditioṅ where the ( −1 ) terms are uniform on . Here, we used (59), (62), and (65) on the second line. The jump contour for ( ; , ) is therefore exactly the same as that for ( ; , ); see Figure 25. From these considerations, we see that uniformly for ( , ) in compact subsets of (ℂ ⧵ ) × ℂ, ( ; , ) satisfies the conditions of a small-norm Riemann-Hilbert problem for | | sufficiently large, and the unique solution satisfies ( ; , and therefore from (33) Now all three factors of ( ; , ) tend to as → ∞ buṫ out ( ; , ) is also diagonal, where in the second line we used (64) and ,1 ( , where in the third line we used (64) and in the fourth line we used (0; , ) = + ( −1 ). Using these results in (69) then gives the asymptotic formula (22) and completes the proof of Theorem 1.
4. ASYMPTOTICS OF ( + ; ) FOR ∈ AND ∈ ℂ ⧵ (ℤ + 1 2 ) To study ( ; ) for values of corresponding to the interior of , we wish to capture two different effects: (i) the rapid oscillation visible in plots showing a locally regular pattern of poles and zeros on a microscopic length scale Δ ∼ 1 and (ii) the gradual modulation of this pattern over macroscopic length scales Δ ∼ . To separate these scales, we write = + as described in Section 1.2. As mentioned in Remark 2, considering ( + ; ) as a function of for fixed ∈ captures the microscopic behavior of , while setting = 0 and considering ( ; ) as a function of captures instead the macroscopic behavior of . A similar approach to the rational solutions of the Painlevé-II equation was taken in [7]. In this section we will develop an approximation of ( + ; ) that depends not on the combination + but rather separately on and in such a way as to explicitly separate these scales. In particular, it will turn out that the approximation is meromorphic in for each fixed but generally is not analytic at all in . 4.1. Spectral curves satisfying the Boutroux integral conditions for ∈ . We tie the spectral curve to the value of the macroscopic coordinate and compensate for nonzero values of the microscopic coordinate later in the construction of a parametrix.

Solving the Boutroux integral conditions for small.
To construct a -function for small, we assume that the spectral curve corresponds to a polynomial ( ; , ) with four distinct roots. We write in polar form as = e i and we write in the form = ̃ . For > 0 we may divide the equations (39) through by √ and consider instead the renormalized Boutroux integral conditions Note that if̃ ∶= Re(̃ ) and̃ ∶= Im(̃ ), then just as in (42) one has that which is nonzero as long as̃ (see the algebraic relation (73)) has distinct branch points on the Riemann sphere of the -plane, see [13, Chapter II, Corollary 1]. We now first set = 0 and attempt to determinẽ as a function of . It is convenient to then reduce the cycle integrals in (72) to contour integrals connecting pairs of branch points in the finite -plane, and since when = 0 the differential̃ d has a double pole with zero residue (in an appropriate local coordinate) at the branch point = 0 we can integrate by parts to transfer "half" of the double pole to each of the finite nonzero roots of̃ 2 which (again in appropriate local coordinates) are simple zeros of̃ d . In this way we obtain conditions equivalent to (72) for = 0 involving a differential that is holomorphic at all three branch points in the finite -plane. These conditions are the following: .
The desired simplification is then that the cycle integrals in (75) over and may be replaced (up to a harmless factor of 2) by path integrals from = 0 to the two roots of the quadratic 2 − 2ĩ + 1 respectively.
If e i = 1, we may solve (75) in this simplified form by assuming̃ to be real and positive. Indeed, then the roots of 2 − 2ĩ + 1 are the values = i(̃ ± √̃ 2 + 1) which lie on the positive and negative imaginary axes. It is easy to see that when = 0,̃ 2 0 > 0 holds for purely imaginary between = i(̃ − √̃ 2 + 1) and = 0. Therefore it is immediate that The remaining Boutroux integral condition then reduces under the hypotheses = 0 and̃ > 0 to a purely real-valued integral condition oñ : Obviously lim̃ ↓0 (̃ ) exists and the limit is positive. Also, by rescaling =̃ , and clearly the first term is the dominant one so (̃ ) < 0 for large positivẽ . Also, by direct calculation, so there exists a unique simple root̃ 0 > 0 of (̃ ). Numerical computation shows that̃ 0 ≈ 0.860437.
If e i = −1, we can invoke the symmetry ↦ − and̃ ↦ −̃ of̃ 2 0 to deduce that the equations (75) hold for = −̃ 0 ≈ −0.860437. When = 0, the elliptic curve given by (73) has distinct branch points on the Riemann sphere unless̃ = ±i, and hence the Jacobian (74) of the equations (75) is nonzero for e i = ±1. The solution of the = 0 system can therefore be continued to other values of e i until the conditioñ ≠ ±i is violated. It is easy to check that̃ = ±i is consistent with (75) only for e i = ∓i. Therefore the solutions of the = 0 system (75) obtained for e i = ±1 can be uniquely continued by the implicit function theorem to fill out an infinitesimal circle surrounding the origin = 0 with the possible exception of its intersection with the imaginary axis. Fixing any phase factor e i ≠ ±i, we can then continue the solution of the full (rescaled) system (72) to small > 0 (in fact, also for < 0, although the solution is not relevant), and the radial continuation can only be obstructed if branch points collide.

4.1.2.
Degenerate spectral curves satisfying the Boutroux integral conditions. The only possible values of ∈ ℂ for which all four roots of ( ; , ) coincide are = ± 1 2 i, which lie on the boundary of . For all ∈ ℂ it is possible to have either a pair of distinct double roots or a double root and two simple roots, provided is appropriately chosen as a function of . We will now show that these degenerate configurations are inconsistent with the Boutroux integral conditions (39), which have to be interpreted in a limiting sense, provided that lies in the interior of but does not also lie on the imaginary axis.
Consider first a nearly degenerate configuration of roots in which two simple roots of are very close to a point = and two reciprocal simple roots are very close to = −1 . Then we may choose the cycle to encircle the pair of roots near, say, = . As the spectral curve degenerates with the cycle fixed, we may observe that becomes in the limit an analytic function of in the interior of and therefore ∮ d → 0 and hence Re(∮ d ) → 0, so one of the Boutroux integral conditions is automatically satisfied in the limit. The cycle should then be chosen to connect the small branch cut near = with the small reciprocal branch cut near = −1 . In the limit that the spectral curve degenerates and 2 becomes a perfect square, the second Boutroux integral condition becomes where + −1 = i −1 . The condition on ∈ ℂ that this quantity vanishes is precisely that either ∈ or lies on the imaginary axis outside of . Therefore the Boutroux conditions cannot be satisfied by such a degenerate spectral curve if is in the interior of .
Next consider a nearly degenerate configuration in which a pair of simple roots of lie very close to = ±1 and another pair of reciprocal simple roots tend to distinct reciprocal limits satisfying whose sum is 2i −1 ∓ 2. Again taking to surround the coalescing pair of roots shows that Re(∮ d ) → 0 in the limit. Then, in the same limit, up to signs, where ± + ( ± ) −1 = 2i −1 ∓ 2 and ± ( ; ) 2 = ( − ± )( − ( ± ) −1 ) with ± having a branch cut connecting the two roots of ± ( ; ) 2 and, say, ± = + (1) as → ∞. It is easy to show that + ( ) = 0 for on the segment between = 0 and = 1 2 i, and that − ( ) = 0 for on the segment between = 0 and = − 1 2 i. However, neither function ± ( ) vanishes identically, so the equations ± ( ) = 0 define a system of curves in the complex -plane. The only branches of these curves in the interior of lie on the imaginary axis as illustrated in Figure 26. Therefore, continuation along radial paths of the Boutroux conditions from the infinitesimal semicircles about the origin in the right and left half-planes defines a unique spectral curve for each ∈ L ∪ R , recalling that L ( R ) is the part of the interior of in the open left (right) half-plane.

Stokes graph and construction of the -function.
For the rest of Section 4 we will be concerned with the approximation of ( + ; ) for large when ∈ ℂ ⧵ (ℤ + 1 2 ) and is bounded, while ∈ L ∪ R . Actually, due to the exact symmetry (5), it is sufficient to assume that ∈ R , as L is the reflection through the origin of R . Thus we assume for the rest of Section 4 that ∈ R and at the end invoke (5) to extend the results to ∈ L .
Given the Stokes graph, we may lay over the arcs Σ out ( ) and in the complement of the Stokes graph a contour consisting of arcs ∞ ⬔ , 0 ⬔ , ∞ ⬕ , and 0 ⬕ that satisfy the increment-of-argument conditions (6)- (7). There are two topologically distinct cases differentiated by the sign Im( ), as illustrated in Figure 30 for ∈ R with Im( ) > 0 and in Figure 31 for ∈ R with Im( ) < 0. If ∈ R with Im( ) = 0, we may use either configuration and obtain consistent results because as a rational function ( ; ) is single-valued. In the rest of this section, we will for simplicity suppose frequently that ∈ R ⧵ ℝ simply for the convenience of being able to speak of contour as a well-defined notion. The vertices of the Stokes graph on the Riemann sphere are the four roots of ( ; , ( )), each of which has degree 3, and the points {0, ∞}, each of which has degree 2.
The solution of Riemann-Hilbert Problem 1 depends parametrically on = + , and when we consider ∈ R we are introducing a -function that depends on but not on . Therefore, in this setting the analogue of the definition (33) is instead i.e., the matrix related to via (84) will depend on both and as independent parameters.

4.2.1.
The -function and its properties. When ∈ R ⧵ ℝ, the self-intersection point of is identified with the root of ( ; , ( )) adjacent to 0 in the Stokes graph. Therefore, for ∈ R ⧵ ℝ, the arcs 0 ⬕ and 0 ⬔ each connect two distinct vertices of the Stokes graph, while ∞ ⬕ joins three consecutive vertices and ∞ ⬔ joins four consecutive vertices. We break these latter arcs at the intermediate vertices; where the integration is over a counterclockwise-oriented loop surrounding ∞,2 ⬔ . As this loop can be interpreted as one of the homology cycles ( , ) on the Riemann surface of the equation 2 = −4 ( ; , ( )), by the Boutroux conditions (39) we therefore have + ( ; )− − ( ; ) = i 1 where 1 ∈ ℝ is a real constant (independent of ∈ ∞,3 ⬔ , but depending on ∈ R ). Also for ∈ ∞,3 ⬔ we have Re( + ( ; )+ − ( ; )− ( ; )) < 0.

Szegő function.
The Szegő function is a kind of lower-order correction to the -function. Its dual purpose is to remove the weak -dependence from the jump matrices on Σ out ( ) for ( ; , , ) defined in (33) while simultaneously repairing the singularity at the origin captured by the condition that To define the Szegő function, we insist that the boundary values taken by ( ; , ) on the arcs of its jump contour Σ out ( ) ∪ 0 ⬕ are related as follows: • Here log(Γ( 1 2 − )) is an arbitrary value of the (generally complex) logarithm, we recall that log ⬕ ( ) ∶= ln | | + iarg ⬕ ( ), and ⟨log ⬕ ( )⟩ refers to the average of the two boundary values of log ⬕ ( ) taken on ∞,2 ⬕ . Also, ( , ) is a constant to be determined so that ( ; , ) tends to a well-defined limit (∞; , ) as → ∞. Writing ( ; , ) = ( ; ) ( ; , ) and solving for using the Plemelj formula we obtain Note that the coefficient of ( , ) is necessarily nonzero as a complete elliptic integral of the first kind. We note also the identity from which it follows that Since ( ; , ) exhibits negative one-half power singularities at each of the four roots of ( ; , ( )), ( ; , ) is bounded near these points. Near the origin, we have ( ; , ) = −( + 1 2 )log ⬕ ( )+(1), and therefore ( ; , , ) is bounded near = 0.
The jump conditions satisfied by ( ; , , ) on the arcs of when ∈ R ⧵ ℝ are then as follows: Similarly, let Λ ± ⬕ denote lens-shaped domains immediately to the left (+) and right (−) of ∞,2 ⬕ , and define For all other values of for which ( ; , , ) is well-defined, we simply set ( ; , , ) ∶= ( ; , , ). If we denote by Λ ± ⬔ (resp., Λ ± ⬕ ) the arc of the boundary of Λ ± ⬔ (resp., Λ ± ⬕ ) distinct from ∞,2 ⬔ (resp., ∞,2 ⬕ ), but with the same initial and terminal endpoints, then the boundary values taken by ( ; , , ) on these arcs satisfy the jump conditions and + ( ; , , ) = − ( ; , , )i 1 e ( , ) 3 e −i 2 ( ) 3 e −i ( ) 3 , On all remaining arcs of , the boundary values of ( ; , , ) agree with those of ( ; , , ), which are related by the jump conditions (93)-(97). Finally, we note that ↦ ( ; , , ) is analytic for ∈ ℂ ⧵ Σ , where Σ ∶= ∪ Λ ± ⬔ ∪ Λ ± ⬕ , taking continuous boundary values from each component of its domain of analyticity, and satisfies ( ; , , ) → as → ∞. The placement of the arcs of relative to the Stokes graph of now ensures that all jump matrices converge exponentially fast to the identity as → +∞ with the exception of those on the arcs ∞,2 ⬔ ∪ ∞,3 ⬔ ∪ ∞,2 ⬕ . The convergence holds uniformly on compact subsets of each open contour arc, as well as uniformly in neighborhoods of = 0 and = ∞. Building in suitable assumptions about the behavior near the four roots of ( ; , ( )), we postulate the following model Riemann-Hilbert problem as an asymptotic description of ( ; , , ) away from these four points.

Jump conditions: The boundary valueṡ out ± ( ) are related on each arc of the jump contour by the following formulae:̇
3. Asymptotics:̇ out ( ) → as → ∞.
The jump diagram for Riemann-Hilbert Problem 3 is illustrated in Figure 32. The solution of this problem (see Section 4.4.2 below) is called the outer parametrix.
Unlike the real-valued quantities 1 ( ) and 2 ( ), ( ) is complex-valued, and it is well-defined because its coefficient is a complete elliptic integral, necessarily nonzero. The boundary values taken by ( ; ) on its jump contour are related by the conditions and We also define a related function ℎ( ; ) by (note that ℎ( ; ) is analytic at = 0 because (0; ) = 1 2 i ), in which ( ) is a constant determined uniquely by setting to zero the coefficient of the dominant term proportional to in the Laurent series of ℎ at = ∞, making ℎ(∞; ) well-defined: The analogues of the conditions (112)-(114) for ℎ are and + ( ; , , ) = − ( ; , , )i 1 e ( ( , )+i ( )) 3 e ( ) 3 , The jump condition (122) together with the continuity of the boundary values taken by ( ; , ) on ∞,3 ⬔ from both sides indicates that the domain of analyticity of ( ; , , ) is precisely the "two-cut" contour Σ out ( ) = ∞,2 ⬔ ∪ ∞,2 ⬕ . Let denote a counterclockwise-oriented loop surrounding the cut ∞,2 ⬔ , and define the Abel mapping ( ; ) by where 0 ( ) is the vertex adjacent to ∞ on the Stokes graph of (hence the initial endpoint of ∞,2 ⬔ ). Note that ( ; ) is well-defined because 1∕ ( ; ) is integrable at = ∞. The integral over the corresponding -cycle (in the canonical homology basis determined from ) of the -normalized holomorphic differential that is the integrand of ( ; ) is then given by with the second equality following from (90), we can use (111) to write ( ) in the form It is a general fact [13] that Re( ( )) < 0, which implies that therefore Re( ( )) ≠ 0 unless 1 ( ) = 0. More concretely, by comparing with the Stokes graphs illustrated in Figure 27, it is easy to see that for > 0 in the domain , ( ) is real and strictly negative. The Abel mapping satisfies the following jump conditions: and We now recall the Riemann theta function Θ( , ) defined by the series The function Θ( , ) has simple zeros only, at each of the lattice points = ( + 1 2 )2 i + ( , Then from the jump conditions (128)-(130) and the automorphic properties (132), it is easy to check that ( ; , , ) satisfies the jump conditions: and To construct ( ; , , ) from , we need to remove the pole from the off-diagonal elements of while slightly modifying the jump conditions on Σ out ( ). To this end, we observe that we have the freedom to introduce mild singularities into ( ; , , ) at the four roots of (⋅; , ( )), here denoted 0 ( ) (adjacent to ∞ in the Stokes graph of ), 1 ( ) (adjacent to 0 ( ) in the Stokes graph), 0 ( ) −1 , and 1 ( ) −1 . Let ( ; ) denote the unique function analytic for ∈ Σ out ( ) with ( ; ) → 1 as → ∞ that satisfies where Therefore the product D ( ; ) OD ( ; ) has precisely one simple zero in its domain of definition, namely = ( ), and this value is either a zero of D ( ; ) or OD ( ; ) but not both. In the case that > 0, the roots of ( ; , ( )) lie on the imaginary axis with 1 < | 1 ( )| < | 0 ( )|. It is easy to check that ( ; ) is positive on the imaginary axis excluding the jump contour Σ out ( ), which also implies that D ( ; ) > 0 for such . The inequality 1 < | 1 ( )| < | 0 ( )| implies that ( ) is negative imaginary, and that | ( )| > | 1 ( ) −1 |. Thus ( ) lies below both intervals of the jump contour Σ out ( ) on the imaginary axis, and hence D ( ( ); ) > 0. It therefore follows that for > 0, ( ) is a zero of OD ( ; ). This will remain so as varies in R so long as ( ) does not pass through either arc of Σ out ( ). We proceed under the assumption that = ( ) is a simple zero of OD ( ; ), and indicate below how the procedure should be modified if ( ) should ever intersect Σ out ( ), a possibility which is difficult to rule out analytically, although we have never observed it numerically.
In order to construct the required function, let ( ± ( )) ∶= ± ( ; ) define properly as a function on Γ, and consider the function ∶ Γ → ℂ given by With the above choice of it is also clear that ( ) = 2 ( ) + (1) as → − (∞), so has a simple pole at = − (∞). Given these choices and the divisor parameter ∈ ℂ, upon taking a generic value of , ( ) will have simple poles at both = + ( ) and = − ( ). We may obviously choose uniquely such that ( ) is holomorphic at = + ( ): With , , determined for arbitrary fixed , there is no additional parameter available in the form (147) to ensure that ( + ( )) = 0, a fact that is consistent with the Riemann-Roch argument given above. So we take the point of view that should be viewed as the additional parameter needed to guarantee that ( + ( )) = 0. Indeed, for this to be the case, the derivative with respect to of the numerator in (147) should vanish at = + ( ); we therefore require: where ( ) is given by (143). These are precisely the three values of ∈ ℂ for which the divisor = + (∞)+ + ( )− − (∞) − − ( ) is special in the setting of the Riemann-Roch theorem. Selecting the desired solution = ( ), it remains only to confirm that (150) holds without squaring both sides. But, since −2 is the sum of roots of ( ; , ( )), from (19) we can also write = −i −1 , and then since when > 0 we know that the branch cuts of ( ; ) lie on opposite halves of the imaginary axis and ( ) lies on the imaginary axis below both cuts, it follows that both sides of (150) are negative imaginary for = ( ) and > 0. The persistence of (150) for = ( ) as varies within R follows by analytic continuation, with the re-definition of ( ( ); ) as described in the last paragraph of Section 4.4.2, should ( ) pass through Σ out ( ) as both move in the complex -plane.  Note that for fixed and = 0, the two conditions in (154) bound within R by a distance from the boundary and also bound away from the points of the Malgrange divisor by a distance proportional to , that is, an arbitrarily small fixed fraction of the spacing between the points of the divisor. .
Rather than try to deal directly with the explicit formula (25), we argue indirectly from the conditions of Riemann-Hilbert Problem 3. We first observe that the outer parametriẋ out ( ; , , ) satisfies a simple algebraic equation. Indeed, it is straightforward to check that the matrix is an entire function; its continuous boundary values match along the three arcs of the jump contour of anḋ out , and it is clearly bounded near the four roots of 2 , hence analyticity in the whole complex -plane follows by Morera's theorem. Moreover, since Liouville's theorem shows that ( ) is a quadratic matrix polynomial in . Using the expansioṅ out ( ; , , ) = where Also, using and the expansioṅ out ( ; , , ) =̇ 0 ,0 ( , , ) +̇ 0 ,1 ( , , ) + ( 2 ) as → 0 gives Therefore ( ) is the quadratic matrix polynomial where, suppressing explicit dependence on the parameters ∈ and ∈ ℂ, Comparing the constant terms between the expansions (161) and (164) yields the identity where ∞ is given by (162), and comparing the terms proportional to in the same expansions yields where 0 1 is given by (165). Since 2 3 = , it is also clear from (159) that the square of the matrix polynomial ( ) is a multiple of the identity, i.e., a specific scalar polynomial: where is the quartic in (19). On the other hand, calculating the square directly from (166) gives We note that the coefficient of 2 here is actually independent of , since according to (19) it is given by = ( ), but the above expression is more useful in the context of the present discussion.
On the other hand, one may observe that the matrix ( ; ) ∶=̇ out ( ; , , )e i ( ) 3 ∕2 satisfies jump conditions that are independent of ∈ ℂ, and therefore −1 is a function of analytic except possibly at = 0 where has essential singularities. By expansion for large and small and Liouville's theorem, it follows that −1 is a Laurent polynomial: Therefore, the outer parametrix out ( ; , , ) itself satisfies the differential equation Substituting the large-expansion oḟ out ( ; , , ) yields an infinite hierarchy of differential equations on the expansion coefficient matrices, the first member of which is Using the off-diagonal part of the identity (170) we can eliminate the commutator [ 3 ,̇ ∞ ,2 ], and therefore (178) implies that ḋ ∞ where (⋅) D denotes the diagonal part of a matrix. Taking the commutator of this equation with 3 then yields Similarly, substituting into (177) the small-expansion oḟ out ( ; , , ) and taking just the leading (constant) term gives the differential equation Multiplying the identity (171) on the right bẏ 0 ,0 allowṡ 0 ,1 to be eliminated from the right-hand side of the above differential equation, leading to ḋ 0 ,0 This identity allows us to compute the derivative of ( ). Using also ( ) 2 = yields the differential equation The differential equations (180) Therefore, using (169) to eliminate ( ) 2 , we find that . (188) Finally, eliminating ( ) using (174) yields the differential equation (19). Together with the fact that the four roots of ( ; , ( )) are distinct by choice of ( ) satisfying the Boutroux conditions (39) on R , this proves the first statement of Theorem 2.
4.5. Airy-type parametrices. Local parametrices for the matrix ( ; , , ) are needed in neighborhoods of each of the four roots of ( ; , ( )), = 0 , 1 , −1 1 , −1 0 , where we recall that by definition 0 is adjacent to ∞ and 1 is adjacent to 0 on the Stokes graph of ∈ R ⧵ ℝ. Centering a disk of sufficiently small radius independent of at each of these points, a conformal map = ( ) can be defined in each disk as indicated in Table 1. As indicated arg( ) = − 2 in this table, we assume that certain contours near −1 0 are fused together within the corresponding disk, and that all contours are locally arranged to lie along straight rays in the -plane emanating from the origin. Locally, the jump contours divide the -plane into four sectors: where ( ) is a piecewise-constant matrix defined in the four sectors of each disk as indicated in Table 2. Note that the Boutroux conditions ∈ ℝ, = 1, 2, imply that ( ) is uniformly bounded on compact sets with respect to ∈ ℂ and for arbitrary ∈ ℤ ≥0 . The jump conditions satisfied by ( ) in each case are most conveniently written in terms  , arg( ) = ± 2 3 , where in each case the boundary values of are defined with respect to orientation in the direction of increasing real part of , and where all powers of are principal branches. We may make a similar transformation of the outer parametrix, noting that in each disk the matriẋ out ( ) ∶=̇ out ( ; , )e i ( ) 3 ∕2 e − ( ; , is analytic except for arg(− ) = 0 where it satisfies exactly the same jump condition as does ( ). This fact, along with the fact that the matrix elements oḟ out ( ) blow up at ( ) = 0 as negative one-fourth powers, implies thaṫ out ( ) can be written in the forṁ out ( ) = ( ; , , ) ( ) 3 ∕4 = ( ; , , ) − 3 ∕6 3 ∕4 , where ( ; , , ) is a function of that is analytic in the disk in question and uniformly bounded with respect to in compact subsets of ℂ ⧵ (ℤ + 1 2 ) and ∈ ℤ ≥0 , provided | | ≤ for some > 0 and satisfy conditions such as enumerated in Lemma 2. Noting that the boundary of each disk corresponds to proportional to 2∕3 , we wish to model the matrix function ( ) by something that satisfies the jump conditions (191) exactly and that matches with the terms 3 ∕4 coming from the outer parametrix when is large. We are thus led to the the following model Riemann-Hilbert problem. , arg( ) = ± 2 3 ,

Riemann-Hilbert
3. Asymptotics: This problem will be solved in all details in Appendix B, where it will be shown that ( ) −1 − 3 ∕4 has a complete asymptotic expansion in descending integer powers of as → ∞, with the dominant terms being given by In each disk we then build a local approximation of ( ; , , ) by multiplying on the left by the holomorphic prefactor ( ; , , ) − 3 ∕6 and on the right by the piecewise-analytic substitution relating ( ; , , ) and where ( ) is the conformal map associated with the disk via Table 1, ( ) is the unimodular transformation matrix given in Table 2, and ( ; , , ) is associated with the outer parametrix and the disk in question via (192)-(193).

Error analysis and proof of Theorem 2.
Let Σ denote the jump contour for the matrix function ( ; , , ), which consists of the contour augmented with the lens boundaries Λ ± ⬔ and Λ ± ⬕ . The global parametrix denoteḋ ( ; , , ) is defined aṡ out ( ; , , ) when lies outside of all four disks, but instead aṡ in ( ; , , ) within each disk (the precise definition is different in each disk as explained in Section 4.5). We wish to compare the global parametrix with the (unknown) matrix function ( ; , , ), so we introduce the error matrix ( ; , , ) defined by ( ; , , ) ∶= ( ; , , )̇ ( ; , , ) −1 . The maximal domain of analyticity of ( ; , , ) is determined from those of the two factors; therefore ( ; , , ) is analytic in except along a jump contour consisting of (i) the part of Σ lying outside of all four disks and (ii) the boundaries of all four disks. That ( ; , , ) can be taken to be an analytic function in the interior of each disk follows from the fact that the inner parametriceṡ in ( ; , , ) satisfy exactly the same jump conditions locally as does ( ; , , ) and an argument based on Morera's theorem. The jump contour for ( ; , , ) corresponding to the Stokes graph shown in Figure 30 is shown in Figure 33.
(205) One way to parametrize points within the domain R is by choosing to set = 0 and thus = where ranges over R . Using this parametrization, we can give the following. Proof of Theorem 3. Given and , for each choice of sign ±, the conditions (204) (resp., (205)) for = 0 define a network of two families of curves whose common intersections locate the zeros (resp., singularities) oḟ ( , 0; ) on L ∪ R . Given ∈ ℂ ⧵ (ℤ + 1 2 ) it is particularly interesting to consider how the curves depend on ∈ ℤ large and positive. For this, we observe that the only dependence on enters through ( , 0, ); substituting from (127) The simplified formulae (206)-(209) show that when is large, the quantization conditions (204)-(205) determine a locally (with respect to ) uniform tiling of the -plane by parallelograms each of which has area (measured in the -coordinate) ◊ ( )(1 + (1)) as → +∞, where see Figure 34. By working in the -plane rather than the -plane, one can see that the area ◊ ( ) is also proportional by a factor of 2 to the Jacobian determinant (42). For each choice of sign ±, one associates via (204) (resp., (205)) FIGURE 34. The local tiling of the -plane by parallelograms of area ◊ ( ).
exactly one zero (resp., pole) oḟ ( , 0; ) with each parallelogram. Hence the densities (per unit -area) of zeros and poles are exactly the same and are given by 2 ( )(1 + (1)) as → +∞, where Note that since 1 ( ) and 2 ( ) are functions independent of ∈ ℂ ⧵ (ℤ + 1 2 ), the same is true of ( ). The density ( ) is a smooth nonnegative function on R , but it vanishes on R ⧵ {0} and blows up as → 0. To prove the former, we may use the fact that ( ) is inversely-proportional to the Jacobian determinant (42), which blows up as and lim ↓0 −1∕2 ( e i ) = ie i 4 ∫ Therefore, the following limit exists: The absolute value of this limit is the quantity ℎ( ) referred to in the statement of Theorem 3.

THE SPECIAL CASE OF
As in Section 3, in this section we study Riemann-Hilbert Problem 1 under the substitution = , i.e., we set = 0.
Here the branch cuts of the two square-root factors emanate to the left from the corresponding roots ± 1 2 i, so the righthand side is analytic in the interior domain with the possible exception of a simple pole at = 0. This particular continuation into through the open arc ∞ ⬔ ⧵ {± 1 2 i} is precisely the function 0 ⬕ ( ), a function that has the arc 0 ⬕ ⊂ as its branch cut. Since (± 1 2 i) 1∕2 = 2 −1∕2 e ±i ∕4 , it is easy to check that Res =0 0 ⬕ ( ) = 0, so 0 ⬕ ( ) is analytic throughout the interior of . We conclude that if = 1 2 + , = 0, 1, 2, 3, … , the asymptotic formula (22) in which ( ) is simply replaced by 0 ⬕ ( ), is valid both for ∈ ℂ ⧵ as well as throughout the maximal domain of analyticity for 0 ⬕ ( ), namely ∈ ℂ ⧵ 0 ⬕ . Likewise, the continuation of ( ) through its branch cut from the left can be written as which is precisely the branch ∞ ⬔ ( ) defined as a meromorphic function on the maximal domain ∈ ℂ ⧵ ∞ ⬔ , the only singularity of which is a simple pole at the origin = 0.

5.2.
Asymptotic behavior of ( ; ) for near the distinguished eyebrow. Proof of Theorem 5. While to describe the asymptotic behavior of ( ; ) for = −( 1 2 + ) with ∈ ℤ ≥0 and bounded away from the eyebrow ∞ ⬔ it was useful to introduce the analytic continuation ∞ ⬔ ( ) of ( ) from a neighborhood of = ∞ to the maximal domain ℂ ⧵ ∞ ⬔ , for near ∞ ⬔ it is better to denote the two critical points of ( ; ) as ( ) and ( ) −1 , both of which are analytic functions on all proper sub-arcs of the eyebrow ∞ ⬔ . We consider the matrix ( ; , ) with the simplest choice of -function, namely ( ) ≡ 0, which will treat the two critical points more symmetrically, as turns out to be appropriate for near the eyebrow ∞ ⬔ . It is then convenient to reformulate the Riemann-Hilbert conditions on ( ; , ) in the special case that = −( 1 2 + ) for ∈ ℤ ≥0 . Since the jump on ∞ ⬕ ∪ 0 ⬕ reduces to the identity in this case (see (11) 3. Asymptotics: ( ) ( ; ) → as → ∞ and ( ) ( ; ) 3 has a well-defined limit as → 0.

5.2.1.
Motivation: the special case of = 0. When = 0, Riemann-Hilbert Problem 5 reduces from a multiplicative matrix problem to an additive scalar problem for the 12-entry, and the explicit solution is obtained from the Plemelj formula: Since ( ; , − 1 2 ) = (0) ( ; ), applying (15) gives the exact result The large-asymptotic behavior of the rational solution ( ; − 1 2 ) is therefore reduced to the classical saddle-point expansion of two related contour integrals. When is close to the eyebrow ∞ ⬔ , Re( ( ( ); )) ≈ 0, so the landscape of Re(− ( ; )) in the -plane is similar to that shown in the central panels of Figure 21, except in small neighborhoods of the two critical points = ( ), ( ) −1 . In particular, for bounded away from these two points, the contour = ∞ ⬔ ∪ 0 ⬔ lies entirely in the red-shaded domain and hence Re(− ( ; )) < 0 holds. This makes the corresponding contributions to the integrands in the numerator and denominator of (227) exponentially small by comparison with the contributions from neighborhoods of the two saddle points. In a sense, this classical saddle point analysis can be embedded in a more general scheme that applies to Riemann-Hilbert Problem 5 also for = 1, 2, 3, … .
Remark 5. In our previous paper on the subject of rational solutions of the Painlevé-III equation [4] we observed that when ∈ ℤ + 1 2 it is possible to reduce Riemann-Hilbert Problem 1 to a linear algebraic Hankel system of dimension independent of in which the coefficients are contour integrals amenable to the classical method of steepest descent when is large such as those just considered above. We originally thought that these Hankel systems would provide the most efficient approach to the detailed analysis of ( ; ) for half-integral , but it turns out that an approach based on more modern techniques of steepest descent for Riemann-Hilbert problems is more effective. We develop this approach in the following paragraphs.

5.2.2.
Modified outer parametrix. The same argument that focuses the contour integrals in (227) on the critical points serves more generally to make the jump matrix in (225) an exponentially small perturbation of the identity matrix except in neighborhoods of the critical points, which in turn suggests approximating ( ) ( ; ) with a single-valued analytic function built to satisfy the required asymptotic conditions as → ∞ and → 0. Thus, given ∈ ℤ ≥0 and nonnegative integers 1 and 2 such that we define an outer parametrix for ( ) ( ; ) by the formulȧ This function is analytic for ∈ ℂ ⧵ {0, ( ), ( ) −1 }, and satisfies the required asymptotic conditions in the sense thaṫ out,( 1 , 2 ) ( ; ) → as → ∞ and thaṫ out,( 1 , 2 ) ( ; ) 3 is analytic at = 0. The singularities in the outer parametrix at the critical points = ( ), ( ) −1 are needed to balance the local behavior of ( ) ( ; ) which we turn to approximating next.

5.2.3.
Inner parametrices based on Hermite polynomials. As the tubular neighborhood containing excludes the branch points = ± 1 2 i, the two critical points remain distinct and hence simple, and both are analytic and nonvanishing functions of ∈ . To set up a uniform treatment of the two critical points, we may also refer to the critical points as Let be simply-connected neighborhoods of ( ), = 1, 2, respectively, and assume that these neighborhoods are sufficiently small but independent of . Exploiting the fact that both critical points of are simple, we conformally map to a neighborhood of the origin via a conformal mapping ↦ ( ; ), where ( ; ) − ( ( ); ) = ( ; ) 2 , ∈ , = 1, 2.
In this equation we make sure to choose branches of log( ) in so that the left-hand side is a well-defined analytic function of that vanishes to second order at the critical point = ( ). For small enough , this relation defines ( ; ) as a conformal mapping up to a sign, which we select such that (possibly after some local adjustment of near the critical points) the image of the oriented arc ∩ is a real interval containing = 0 traversed in the direction of increasing . We will need the precise value of ′ ( ( ); ), and by implicit differentiation one finds Likewise, the function ↦ √ 2 −1∕2 ∞ ∕ ! admits analytic continuation from ∩ to all of , and this function is non-vanishing on (taken sufficiently small but independent of ). Therefore, it has an analytic and non-vanishing square root, which we denote by ( ), = 1, 2. Then, using (225) the jump condition for the modified matrix To define appropriate solutions of these jump conditions within the neighborhoods yielding inner parametrices matching well onto the outer parametrix when ∈ , we need to take into account the final factor on the right-hand side of (231). Thus writing = 1∕2 ( ; ), we arrive at the following model Riemann-Hilbert problem.
To compare ( ) ( ; ) with its parametrix, we define two types of comparison matrices by of Re( ( ; )) shown in these plots resembles, at least for not too close to 1 or 2 , that illustrated in the central panels of Figure 21. Therefore, when is close to the eyebrow ∞ ⬔ , if the tubular neighborhood is taken to be sufficiently thin (by choosing 2 sufficiently small in (32)) given the domains 1 and 2 , the jump condition satisfied by ( 1 , 2 , ) ( ; ) on the arcs of exterior to the latter domains has the form (because all red contours lie strictly within the pink-shaded region) with the convergence being in the sense for every and holding uniformly for ∈ . Therefore, the essential jump conditions for ( 1 , 2 , ) ( ; ) are those across the domain boundaries 1 and 2 . Taking these to be oriented in the hold, then (253) becomes By varying the index = 1, 2 as well as the choice of non-negative integers 1 + 2 = , the above inequalities (257),

Modeling of comparison matrices.
To determine the asymptotic behavior as → +∞ of the various types of comparison matrices, it now becomes necessary to model the leading terms of the jump matrices, which generally do not decay to the identity on the domain boundaries 1 and 2 , but that are guaranteed to be bounded by association of ∈ with the appropriate indices 1 , 2 , and as described above. In Cases I and II a , the dominant terms in the jump matrices on 1 and 2 are both upper triangular matrices, while in Case II b one jump matrix is upper triangular and the other is lower triangular. This situation requires two different types of parametrices, which we formulate as Riemann-Hilbert problems here.
This problem always has a unique solution, which may be sought in the forṁ The conditions of Riemann-Hilbert Problem 7 descend to the conditions that the scalar functioṅ ( ) be analytic for ∈ ℂ ⧵ ( 1 ∪ 2 ) witḣ ( ) → 0 as → ∞, and the jump conditions now take the additive form:̇ + ( ) = − ( ) + ( ) holds for ∈ , = 1, 2. It follows thaṫ ( ) is given by the Plemelj formulȧ The integrals may be evaluated explicitly, by residues. In the special case that is exterior to both domains , = 1, 2, we therefore finḋ where Φ denotes the residue of at , = 1, 2. In particular, we see thaṫ in which we havė and assuming clockwise orientation of L , 3. Asymptotics:̇ ( ) → as → ∞.
This problem is a generalization of the one that characterizes the soliton solutions of AKNS systems [2]. Unlike Riemann-Hilbert Problem 7 this problem is only conditionally solvable. Letting Φ U denote the residue of U at U , and Φ L that of L at L , this problem has a unique solution if and only if Δ ∶= Φ U Φ L + ( U − L ) 2 ≠ 0.
The solution is a rational function in the domain exterior to U ∪ L : This formula determineṡ ( ) in the domains U and L by the jump conditions; Laurent expansion of the interior boundary valuė − ( ) shows in each case that its only singularity is removable. Moreover, (270) shows that expansions of the form (265) again hold whenever the solution exists, in whicḣ Applying similar arguments to express the ratio of residues appearing in (278) in terms of integer powers of ( ) giveṡ Comparing (288) and (291), we observe that the approximate formulȧ is the same over the whole range − 1 2 ( − 2 + We observe that (285) and (294) agree upon replacing with + 1 in the former.

Detailed asymptotics for poles and zeros. Proofs of Corollary 4 and Theorem 6.
Proof of Corollary 4. Each of the formulae foṙ described in Section 5.2.7 is a different meromorphic function of whose accuracy as an approximation of ( ; −( 1 2 + )) holds in an absolute sense for in a certain curvilinear strip roughly parallel to the eyebrow ∞ ⬔ and of width proportional to −1 ln( ). The absolute accuracy of the approximation depends on the assumption that is bounded away from each pole and zero oḟ by a distance proportional to −1 by an arbitrarily small constant. It is easy to see that this distance is an arbitrarily small fraction of the spacing between nearest poles or zeros oḟ . This allows one to compute the index (winding number) of ( ; −( 1 2 + )) about a small circle containing just one pole or zero oḟ and hence deduce that the index is −1 or 1 respectively.