Connection problem for the tau-function of the Sine-Gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach
CConnection problem for the tau-function of theSine-Gordon reduction of Painlev´e-III equation via theRiemann-Hilbert approach.
A. Its
Department of Mathematical Sciences,Indiana University – Purdue University IndianapolisIndianapolis, IN 46202-3216, USA
A. Prokhorov
Department of Mathematical Sciences,Indiana University – Purdue University IndianapolisIndianapolis, IN 46202-3216, USA
Abstract.
We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in thelarge 𝑥 asymptotics of the Painlev´e III tau-function. Our result proves the conjectural formulafor this pre-factor obtained recently by O. Lisovyy, Y. Tykhyy, and the first co-author withthe help of the recently discovered connection of the Painlev´e tau-functions with the Virasoroconformal blocks. Our approach does not use this connection, and it is based on the Riemann-Hilbert method. We consider the particular case of the third Painlev´e equation, which is a radial-symmetricreduction of the elliptic sine-Gordon equation 𝑢 𝑥𝑥 + 𝑢 𝑥 𝑥 + sin 𝑢 = 0 . (1)Starting from the pioneering works [1], [15] on the Ising model, this equation has been playingan increasingly important role, as a “nonlinear Bessel function”, in a growing number of physicalapplications (see e.g., [4] and references therein). Apparently, the first appearance of equation(1) in the physical applications should be credited to work of J. M. Myers [16].Equation (1) can be written as a (non-autonomous) Hamiltonian system, 𝑑𝑢𝑑𝑥 = 𝜕 ℋ 𝜕𝑣 , 𝑑𝑣𝑑𝑥 = − 𝜕 ℋ 𝜕𝑢 , a r X i v : . [ m a t h - ph ] N ov n the phase space R = { ( 𝑢, 𝑣 ) } equipped with the canonical symplectic structure,Ω = 𝑑𝑣 ∧ 𝑑𝑢. (2)The Hamiltonian ℋ is given by the formula ℋ = 𝑣 𝑥 − 𝑥 cos 𝑢. We are concerned with the global asymptotic analysis of the 𝜏 - function corresponding to thePainlev´e III equation (1) which is defined according to the equation (see [9], [19]), 𝑑 ln 𝜏𝑑𝑥 = − ℋ . (3)In fact, it is this tau-function (evaluated for a special family of solutions of equation (1)) thatplayed a key role in the above mentioned Barouch-McCoy-Tracy-Wu theory, and, since then, ithas appeared in many problems of statistical mechanics and quantum field theory. Let us nowremind some of the basic known facts about the asymptotics of the solutions of equation (1).We refer the reader to monograph [4] for more details and for the history of the question.Equation (1) possesses a two-parameter family of solutions characterized by the followingbehavior at 𝑥 = 0, 𝑢 ( 𝑥 ) = 𝛼 ln 𝑥 + 𝛽 + 𝑂 (︀ 𝑥 −|ℑ 𝛼 | )︀ , 𝑥 → , (4)where the complex numbers 𝛼 ∈ C , |ℑ 𝛼 | < 𝛽 ∈ C can be taken as parameters - theCauchy data, of the solution 𝑢 ( 𝑥 ) ≡ 𝑢 ( 𝑥 | 𝛼, 𝛽 ). The behavior of the solution 𝑢 ( 𝑥 | 𝛼, 𝛽 ) as 𝑥 → + ∞ , is known. For an open set in the space of parameters 𝛼 , 𝛽 , which we will describelater, the large 𝑥 behavior of 𝑢 ( 𝑥 | 𝛼, 𝛽 ) is oscillatory, and it is given by the formulae, 𝑢 ( 𝑥 ) = 𝑏 + 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − / (︂ 𝑂 (︂ 𝑥 )︂)︂ + 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − / (︂ 𝑂 (︂ 𝑥 )︂)︂ ++ 𝑂 (︀ 𝑥 |ℑ 𝜈 |− / )︀ (mod 2 𝜋 ) , 𝑥 → ∞ , (5)where 𝜈 = − 𝑏 + 𝑏 − , |ℑ 𝜈 | < / . (6)The asymptotic parameters at infinity - the complex amplitudes 𝑏 ± , can be, in fact, expressed interms of the Cauchy data 𝛼 , 𝛽 , and the condition |ℑ 𝜈 | < / connection formulae were obtained in 1985 by V. Yu. Novokshenov[18] (see also: [10] and [11]), and they are given by the equations, 𝑒 𝜋𝜈 = sin 2 𝜋𝜂 sin 2 𝜋𝜎 , 𝑏 ± = − 𝑒 𝜋𝜈 ∓ 𝑖𝜋 ± 𝑖𝜈 √ 𝜋 Γ(1 ∓ 𝑖𝜈 ) sin 2 𝜋 ( 𝜎 ∓ 𝜂 )sin 2 𝜋𝜂 , (7)where 𝜎 := 14 + 𝑖 𝛼, 𝜂 := 14 + 14 𝜋 (︁ 𝛽 + 𝛼 ln 8 )︁ + 𝑖 𝜋 ln Γ (︀ − 𝑖𝛼 )︀ Γ (︀ + 𝑖𝛼 )︀ , (8)2nd Γ( 𝑧 ) is Euler’s Gamma-function. The open set in the space of the Cauchy data 𝛼 , 𝛽 wherethe both asymptotics, (4) and (5) are valid is described by the inequalities (see also Remark 2below),0 < ℜ 𝜎 < ⇐⇒ |ℑ 𝛼 | < , sin 2 𝜋𝜂 ̸ = 0 , ⃒⃒⃒⃒ arg sin 2 𝜋𝜂 sin 2 𝜋𝜎 ⃒⃒⃒⃒ < 𝜋 ⇐⇒ |ℑ 𝜈 | < , (9)where 𝜎 and 𝜂 are understood as functions of 𝛼 and 𝛽 defined in (8). We notice that this setcontains all sufficiently small pairs ( 𝛼, 𝛽 ), all real pairs ( 𝛼, 𝛽 ) such that the corresponding 𝜂 satisfies the inequality 0 < 𝜂 < (mod (1)) and all pure imaginary pairs such that | 𝛼 | <
2. Infact, it is convenient to take 𝜎 and 𝜂 as the independent parameters and think about 𝛼 and 𝛽 as their functions, i.e., 𝛼 = 𝑖 (2 − 𝜎 ) , 𝛽 = − 𝜋 + 4 𝜋𝜂 − 𝑖 (2 − 𝜎 ) ln 8 − 𝑖 ln Γ(1 − 𝜎 )Γ(2 𝜎 ) , (10)where 𝜎 , 𝜂 are the complex numbers satisfying (9). The expressions of the asymptotic param-eters at 𝑥 = ∞ in terms of 𝜎 and 𝜂 have already been presented in (7).The derivation of formulae (7) is based on the Isomonodromy-Riemann-Hilbert Method . Weagain refer the reader to monograph [4] for more details and for general references concerning theconnection problem for Painlev´e equations. In the framework of the Riemann-Hilbert method,the parameters 𝜎 and 𝜂 have an independent important meaning as the monodromy data ofthe auxiliary linear system associated with the third Painlev´e equation. This meaning of theparameters 𝜎 and 𝜂 plays important role in the considerations of this paper, and it will beexplained in detail in the next section.Equations (4) and (5) in turn imply the following behavior at zero and at infinity of thecorresponding tau-function (see also [8]), 𝜏 ( 𝑥 ) = 𝐶 𝑥 − 𝛼 (1 + 𝑜 (1)) , 𝑥 → , (11)and 𝜏 ( 𝑥 ) = 𝐶 ∞ 𝑥 𝜈 𝑒 𝑥 +2 𝜈𝑥 (1 + 𝑜 (1)) , 𝑥 → ∞ . (12)In fact, one can write a complete asymptotic series for the tau-function at both critical pointswhose coefficients are explicit functions of the Cauchy data 𝛼 , 𝛽 or, equivalently, of the mon-odromy data 𝜎 , 𝜂 . The issue which we are concerned with is the evaluation of the ratio 𝐶 ∞ /𝐶 (13)in terms of the initial data 𝛼 , 𝛽 . This can not be done just by using the asymptotic equations(4) - (5) and the connection formulae (7)-(8). Indeed, we are dealing here with the “constant ofintegration” problem. For the special one-parameter family of solutions of equation (1) relatedto the Ising model, this problem was solved by C. Tracy [20] in 1991. This special family isobtained by putting 𝜂 = 0 and 𝜎 ∈ R , < 𝜎 < . (14)3n (10). Zero value of 𝜂 is excluded from set (9) which means that the behavior of this specialfamily at infinity is very different from the oscillatory one given in (5). In fact, all the solutionsfrom this family exponentially approach 𝜋 (mod2 𝜋 ), 𝑢 ( 𝑥 ) − 𝜋 ∼ 𝑖𝜅 √︂ 𝜋 𝑥 − / 𝑒 − 𝑥 , 𝑥 → ∞ , 𝜅 = − 𝜋𝜎. In his calculations, Tracy made use of the existence of an additional Fredholm determinantrepresentation of the tau-function evaluated on the family (14). We are interested in a generic,two-parameter case where there is no such representation. A conjectural answer to the problemhas been produced in [12] with the help of the recently discovered by O. Gamayun, N. Iorgov,and O. Lisovyy connection of the Painlev´e tau-functions with the Virasoro conformal blocks[5], [6]. In this paper we prove the conjecture of [12]. Our main result is the following theorem.
Theorem 1.
Let 𝜎 and 𝜂 be the “monodromy” parameters of the Painlev´e III function 𝑢 ( 𝑥 ) satisfying the inequalities (9). Then the ratio (13) is given by the formula, 𝐶 ∞ 𝐶 = 2 𝑒 − 𝑖 𝜋 𝜋 ( 𝐺 ( )) (2 𝜋 ) 𝑖𝜈 𝜈 + 𝜎 − 𝜎 𝑒 𝜋𝑖 ( 𝜂 − 𝜎𝜂 − 𝜎 +2 𝜂 − 𝜎 ) × Γ(1 − 𝜎 )Γ(2 𝜎 ) (︂ 𝐺 (1 + 𝑖𝜈 ) 𝐺 (1 + 2 𝜎 ) 𝐺 (1 − 𝜎 ) 𝐺 (1 + 𝜎 + 𝜂 + − 𝑖𝜈 ) 𝐺 ( − 𝑖𝜈 − 𝜎 − 𝜂 ) 𝐺 (1 + 𝜎 + 𝜂 + 𝑖𝜈 ) 𝐺 ( 𝑖𝜈 − 𝜎 − 𝜂 ) )︂ , (15) where 𝜈 is defined in (7) and 𝐺 ( 𝑧 ) is the Barnes 𝐺 - function. It should be noticed that in [12] a slightly different definition of the tau-function is used.The exact relation of the constant (15) and the one conjectured in [12] is discussed in the lastsection of the paper.Our proof of Theorem 1 is not based on the conformal block connection. We use theRiemann-Hilbert representation of the third Painlev´e transcendent and the Malgrange-Bertolaextension of the Jimbo-Miwa-Ueno definition of the tau-function.In the course of our proof, we also confirm one of the key observations of [12] that ratio (15)determines the generating function of the canonical transformation of the canonical variablesdetermined by the initial data ( 𝛼, 𝛽 ) to the canonical variables determined by the asymptoticdata ( 𝑏 + , 𝑏 − ) (see the end of Section 5 for more detail). In fact, this Hamiltonian interpretationof the pre-factors in the asymptotics of the Painlev´e tau-functions was first suggested in thework [7] of N. Iorgov, O. Lisovyy and Yu. Tykhyy.The evaluation of ratio (13), which we have made rigorous in this paper, is only one ofa series of highly nontrivial predictions and already established facts which came from theremarkable discovery of Gamayun, Iorgov, and Lisovyy. These other predictions and results,including the key ingredient of the approach of [5] and [6], which is the explicit conformal blockseries representations for the Painlev´e tau-functions, do not yet have their understanding inthe framework of the Riemann-Hilbert method.We shall start the proof of Theorem 1 with the reminding of the Isomonodromy-Riemann-Hilbert formalism for the Painlev´e equation (1) (for more detail see, e.g., [4]).4 The Riemann-Hilbert Representation of the Solutionsof the Sine-Gordon/Painlev´e III Equation
The Riemann-Hilbert problem associated with equation (1) is posed on the oriented contourΓ depicted in Figure 1, and it consists in the finding of a 2 × 𝜆 )which satisfies the following properties. 0 𝑖 − 𝑖 𝐸 − 𝜎 𝐸 − 𝜎 𝑆 ( ∞ )1 𝑆 (0)1 𝑆 (0) − 𝑆 ( ∞ ) − Figure 1: Contour Γ ∙ The function Ψ( 𝜆 ) is analytic on C ∖ { Γ } , it has continuous ± - limits on the contourΓ, and these limits satisfy jump condition Ψ + ( 𝜆 ) = Ψ − ( 𝜆 ) 𝑆 ( 𝜆 ). Here “ + ” denotes theboundary values from the left side of the contour and “ − ” denotes the boundary valuesfrom the right side of the contour. Jump matrix 𝑆 ( 𝜆 ) is piecewise constant, its differentcomponents are indicated in Figure 1, and they are given by the equations, 𝑆 ( ∞ )1 = 𝑆 (0)2 = (︂ 𝑝 + 𝑞 )︂ , 𝑆 ( ∞ )2 = 𝑆 (0)1 = (︂ 𝑝 + 𝑞 )︂ , (16) 𝐸 = 1 √ 𝑝𝑞 (︂ 𝑝 − 𝑞 )︂ , 𝑝, 𝑞 ∈ C , 𝑝𝑞 ̸ = 0 , 𝜎 = (︂ )︂ . (17) ∙ The function Ψ( 𝜆 ) satisfies the following conditions at zero and infinityΨ( 𝜆 ) = 𝑃 ( 𝐼 + 𝑀 (0)1 𝜆 + 𝑂 ( 𝜆 )) 𝑒 − 𝑖𝜆 𝜎 , 𝜆 → , Ψ( 𝜆 ) = (︁ 𝐼 + 𝑀 ( ∞ )1 𝜆 + 𝑂 (︁ 𝜆 )︁)︁ 𝑒 − 𝑖𝑥 𝜆 𝜎 , 𝜆 → ∞ , (18)5here 𝑃 , 𝑀 (0)1 , 𝑀 ( ∞ )1 here are some constant in 𝜆 matrices and 𝜎 = (︂ − )︂ . The Riemann-Hilbert problem is uniquely and meromorphically in 𝑥 solvable for all 𝑝, 𝑞 ∈ C , 𝑝𝑞 ̸ = 0 [17] and the corresponding solution 𝑢 ( 𝑥 ) ≡ 𝑢 ( 𝑥 ; 𝑝, 𝑞 ) of the thirdPainlev´e equation (1) is given by the formula, 𝑢 ( 𝑥 ) = 2 arccos( 𝑃 ) . In fact, the following equation takes place, 𝑃 = (︂ cos( 𝑢 ) − 𝑖 sin( 𝑢 ) − 𝑖 sin( 𝑢 ) cos( 𝑢 ) )︂ ≡ 𝑒 − 𝑖𝑢𝜎 . (19) Remark 1.
The above Riemann-Hilbert setting corresponds to the generic solutions of (1).There is one parameter family of a separatrix solution which is characterized by the followingRiemann-Hilbert data 𝑆 ( ∞ )1 = 𝑆 (0)2 = (︂ 𝜅 )︂ , 𝑆 ( ∞ )2 = 𝑆 (0)1 = (︂ 𝜅 )︂ , 𝐸 = ± 𝑖 (︂ )︂ , 𝜅 ∈ C . This is the family which includes the McCoy-Tracy-Wu solution (14) and which is not consideredin this paper. As it has already been mentioned, the constant problem for this family was solvedin [20].
The parameters 𝑝, 𝑞 ∈ C in (16), (17) are connected with the parameters of asymptotic of 𝑢 ( 𝑥 ) via 𝑝 := − 𝑖 sin 2 𝜋 ( 𝜎 + 𝜂 )sin 2 𝜋𝜂 , 𝑞 := 𝑖 sin 2 𝜋 ( 𝜎 − 𝜂 )sin 2 𝜋𝜂 . Conditions |ℑ 𝛼 | < |ℑ 𝜈 | < / 𝑝 and 𝑞 as 𝑝 + 𝑞 / ∈ ( − 𝑖 ∞ , − 𝑖 ] ∪ [2 𝑖, + 𝑖 ∞ ) , and 𝑝𝑞 / ∈ ( −∞ , − , (20)respectively. Remark 2.
Conditions (20) are the conditions which appear during the asymptotic analysis ofthe Riemann-Hilbert problem. The asymptotic parameter 𝜈 is related to 𝑝 , 𝑞 according to theequation, 𝜈 = − 𝜋 ln(1 + 𝑝𝑞 ) . Also, 𝑝𝑞 = sin 𝜋𝜎 sin 𝜋𝜂 . We restrict ourselves in (9) to the inequality ⃒⃒ arg sin 2 𝜋𝜂 sin 2 𝜋𝜎 ⃒⃒ < 𝜋/ , instead of the inequality ⃒⃒⃒ arg (︀ sin 2 𝜋𝜂 sin 2 𝜋𝜎 )︀ ⃒⃒⃒ < 𝜋 by a technical reason. This means that we actually analyze one of the omponents of the full set of the Cauchy data corresponding to the generic asymptotic behavior(4) and (5). The another component is defined by the condition, ⃒⃒⃒⃒ arg sin 2 𝜋𝜂 sin 2 𝜋𝜎 − 𝜋 ⃒⃒⃒⃒ < 𝜋/ , which in turn implies the following change in formulae (7), 𝑒 𝜋𝜈 = − sin 2 𝜋𝜂 sin 2 𝜋𝜎 . The analysis presented in this paper can be easily extended on this component of initial dataas well. Actually, the 𝜏 -function does not change if we add 𝜋𝑖 to the function 𝑢 ( 𝑥 ) . But 𝜂 isshifted by . So such change of variable allows us to go from one component to another. Function Ψ( 𝜆 ) satisfies system of linear ordinary differential equations 𝑑 Ψ 𝑑𝜆 = 𝐴 ( 𝜆 )Ψ( 𝜆 ) , (21) 𝑑 Ψ 𝑑𝑥 = 𝑈 ( 𝜆 )Ψ( 𝜆 ) ,𝐴 ( 𝜆 ) = − 𝑖𝑥 𝜎 − 𝑖𝑥𝑢 𝑥 𝜎 𝜆 + 𝑃 ( 𝑖𝜎 ) 𝑃 − 𝜆 , (22) 𝑈 ( 𝜆 ) = − 𝑖𝜆𝑥𝜎 − 𝑖𝑢 𝑥 𝜎 . Equation (1) is the compatibility condition for this system and it describes isomonodromicdeformations of the system (21). From this point of view 𝜎 and 𝜂 play role of the monodromydata.We complete this overview of the Riemann-Hilbert formalism for equation (1) by presentingthe general alternative definition of the Jimbo-Miwa-Ueno tau-function in terms of the solutionΨ( 𝜆 ) of the Riemann-Hilbert problem.Define ˆΨ ( ∞ ) ( 𝜆 ) := Ψ( 𝜆 ) 𝑒 𝑖𝑥 𝜆 𝜎 , | 𝜆 | > 𝑅. Then, according to [9] the equation, 𝜔 𝐽𝑀𝑈 = − 𝑟𝑒𝑠 𝜆 = ∞ Tr (︁ ( ˆΨ ( ∞ ) ( 𝜆 )) − ( ˆΨ ( ∞ ) ( 𝜆 )) ′ ( − 𝑖𝜆𝑥 𝜎 ) )︁ 𝑑𝑥, (23)defines the differential form whose antiderivative is the logarithm of tau-function. Actually,from (18) we have thatˆΨ ( ∞ ) ( 𝜆 ) = 𝐼 + 𝑀 ( ∞ )1 𝜆 + 𝑂 (︁ 𝜆 )︁ . (24)Substituting (18) to the equation (21), one can express 𝑀 ( ∞ )1 in terms of 𝑢 ( 𝑥 ) (see [4],[17]). 𝑀 ( ∞ )1 = − 𝑖𝑥𝑢 𝑥 𝑥 𝜎 − 𝑖 (︂ cos 𝑢 − 𝑢 𝑥 )︂ 𝜎 , 𝜎 = (︂ − 𝑖𝑖 )︂ . (25)7ubstituting, in turn, (24)-(25) into (23) yields the relation, 𝜔 𝐽𝑀𝑈 = 𝑑 ln 𝜏 ( 𝑥 ), where 𝑑 ln 𝜏 isdefined in (3). In other words, the tau-function can be alternatively defined as 𝜏 ≡ 𝜏 𝐽𝑀𝑈 ( 𝑥, 𝑝, 𝑞 ) = 𝑒 ∫︀ 𝜔 𝐽𝑀𝑈 . So defined the tau-function is unique up to multiplication by a constant depending on 𝑝 and 𝑞 .The key fact for us is that, following [2], it is possible to extend Jimbo-Miwa-Ueno differentialform (23) on vector fields in 𝑝 and 𝑞 in such a way, that it will remains a closed form. Suchextension will allow us to define tau-function already up to a constant, which does not dependon 𝑝 and 𝑞 . In this section we basically repeat the calculations and the results of Section 5.1 of paper [2]adjusting them to our special case.Put 𝑌 ( 𝜆 ) = Ψ( 𝜆 ) 𝑒 ( 𝑖𝑥 𝜆 + 𝑖𝜆 ) 𝜎 . Denote 𝐺 ( 𝜆 ) the jump matrix for 𝑌 ( 𝜆 ). Following [2],[14], wedefine the Malgrange-Bertola differential form by the equation 𝜔 𝑀𝐵 [ 𝜕 ] = ∫︁ Γ Tr( 𝑌 − − 𝑌 ′− ( 𝜕𝐺 ) 𝐺 − 𝑑𝜆 𝜋𝑖 . (26)Here 𝜕 denotes the vector field in the space of parameters 𝑥, 𝑝, 𝑞 , and the prime denotes deriva-tive with respect to 𝜆 . This differential form was introduced originally by B. Malgrange in [14]for the case when the contour Γ is a circle. M. Bertola in [2] has extended the Malgrange’sdefinition to an arbitrary Riemann-Hilbert setting.Let us establish the connection of this form with Jimbo-Miwa-Ueno form. For the case ofgeneral Riemann-Hilbert problem, the analog of this Lemma was proven in [2]. Lemma 1.
The Malgrange-Bertola differential form, evaluated on the vector fields in parameter 𝑥 , is equal to the Jimbo-Miwa-Ueno form up to a term, depending only on 𝐺 ( 𝜆 ) . 𝜔 𝑀𝐵 [ 𝜕 𝑥 ] = 𝜔 𝐽𝑀𝑈 [ 𝜕 𝑥 ] − ⎡⎣∫︁ Γ Tr (︁ 𝐺 − 𝐺 ′ 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 ⎤⎦ − 𝑥 . (27) Proof.
First, we have 𝐺 ( 𝜆 ) = 𝑒 − ( 𝑖𝑥 𝜆 + 𝑖𝜆 ) 𝜎 𝑆 ( 𝜆 ) 𝑒 ( 𝑖𝑥 𝜆 + 𝑖𝜆 ) 𝜎 , where 𝑆 ( 𝜆 ) is the jump matrix for Ψ( 𝜆 ). Hence, 𝜕 𝑥 𝐺 ( 𝜆 ) 𝐺 − ( 𝜆 ) = − 𝑖𝑥𝜆 𝜎 + 𝑖𝑥𝜆 𝐺 ( 𝜆 ) 𝜎 𝐺 − ( 𝜆 ) . (28)Also, we have that 𝑌 + ( 𝜆 ) = 𝑌 − ( 𝜆 ) 𝐺 ( 𝜆 ) , (29)and, by 𝜆 - differentiation, 𝑌 ′ + ( 𝜆 ) = 𝑌 ′− ( 𝜆 ) 𝐺 ( 𝜆 ) + 𝑌 − ( 𝜆 ) 𝐺 ′ ( 𝜆 ) . (30)8ubstituting (28),(29),(30) in (26) we get 𝜔 𝑀𝐵 [ 𝜕 𝑥 ] = ⎡⎣∫︁ Γ Tr (︁ ( 𝑌 − 𝑌 ′ + − 𝑌 − − 𝑌 ′− ) 𝑖𝑥𝜆 𝜎 − 𝐺 − 𝐺 ′ 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 ⎤⎦ . Let us introduce the notation for the parts of the contour Γ as it is indicated in Figure 2.Γ 𝑖 − 𝑖 Γ Γ Γ Γ Γ Figure 2: Parts of the contour Γ9e have ∫︁ Γ ∪ Γ ∪ Γ Tr (︁ 𝑌 − 𝑌 ′ + 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 = 0 , ∫︁ Γ Tr (︁ 𝑌 − 𝑌 ′ + 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 − ∫︁ Γ ∪ Γ Tr (︁ 𝑌 − − 𝑌 ′− 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 = 0 . We also have 𝑌 ( 𝜆 ) = ˆΨ ( ∞ ) ( 𝜆 ) 𝑒 𝑖𝜆 𝜎 , | 𝜆 | > 𝑅. Hence, ∫︁ Γ ∪ Γ Tr (︁ 𝑌 − 𝑌 ′ + 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 − ∫︁ Γ ∪ Γ ∪ Γ ∪ Γ Tr (︁ 𝑌 − − 𝑌 ′− 𝑖𝑥𝜆 𝜎 )︁ 𝑑𝜆 𝜋𝑖 = res 𝜆 = ∞ Tr (︁ 𝑒 − 𝑖𝜆 𝜎 ( ˆΨ ( ∞ ) ( 𝜆 )) − ( ˆΨ ( ∞ ) ( 𝜆 )) ′ 𝑒 𝑖𝜆 𝜎 𝑖𝑥𝜆 𝜎 − 𝑒 − 𝑖𝜆 𝜎 ( ˆΨ ( ∞ ) ( 𝜆 )) − ( ˆΨ ( ∞ ) ( 𝜆 )) 𝑖𝜆 𝜎 𝑒 𝑖𝜆 𝜎 𝑖𝑥𝜆 𝜎 )︁ = res 𝜆 = ∞ Tr (︁ ( ˆΨ ( ∞ ) ( 𝜆 )) − ( ˆΨ ( ∞ ) ( 𝜆 )) ′ 𝑖𝑥𝜆 𝜎 )︁ − 𝑥 , and (27) follows.This lemma means, that Malgrange-Bertola form is indeed a good candidate for extensionof Jimbo-Miwa-Ueno form. However, there is an additional term, depending only on 𝐺 ( 𝜆 ). Onecan, following again [2], cancel it considering the modified Malgrange-Bertola form 𝜔 = 𝜔 𝑀𝐵 + 𝜃 ,where 𝜃 [ 𝜕 ] = 12 ∫︁ ^Γ Tr( 𝐺 ′ 𝐺 − ( 𝜕𝐺 ) 𝐺 − ) 𝑑𝜆 𝜋𝑖 . In the notations of [2] 𝜔 is the form Ω from Definition 2.2 of [2].We have 𝜔 [ 𝜕 𝑥 ] = 𝜔 𝐽𝑀𝑈 [ 𝜕 𝑥 ] − 𝑥 . Indeed, 𝐺 − 𝐺 ′ 𝑖𝑥𝜆 𝜎 = 𝑖𝑥𝜆 (︂ 𝑖𝑥 − 𝑖𝜆 )︂ ( 𝐼 − 𝐺 − 𝜎 𝐺𝜎 ) , Tr( 𝐺 ′ 𝐺 − ( 𝜕 𝑥 𝐺 ) 𝐺 − ) = 𝑖𝑥𝜆 (︂ 𝑖𝑥 − 𝑖𝜆 )︂ 𝐼 − 𝐺 − 𝜎 𝐺𝜎 ) , and the additional term depending only on 𝐺 ( 𝜆 ) cancels.In the next section we will express the form 𝜔 in terms of the coefficients of the asymptoticexpansions of 𝑌 ( 𝜆 ) at 𝜆 = 0 and at 𝜆 = ∞ . We call this expression a “localization” ofthe original integral formula (26) for the Malgrange-Bertola form. This localized version willsimplify dramatically the further analysis of the form 𝜔 . Let us introduce the function Θ( 𝜆 ) = 𝜕𝑌 ( 𝜆 ) 𝑌 ( 𝜆 ) − , 𝜕 means the differentiation with respect to one of the three parameters, 𝑥 , 𝑝 , or 𝑞 . The 𝜕 -version of equation (30) reads, 𝜕𝑌 + ( 𝜆 ) = 𝜕𝑌 − ( 𝜆 ) 𝐺 ( 𝜆 ) + 𝑌 − ( 𝜆 ) 𝜕𝐺 ( 𝜆 ) . Expressing 𝐺 ( 𝜆 ) from (29) as 𝐺 ( 𝜆 ) = 𝑌 − − ( 𝜆 ) 𝑌 + ( 𝜆 ) we rewrite the last equations as 𝜕𝑌 + ( 𝜆 ) 𝑌 − ( 𝜆 ) = 𝜕𝑌 − ( 𝜆 ) 𝑌 − − ( 𝜆 ) + 𝑌 − ( 𝜆 ) 𝜕𝐺 ( 𝜆 ) 𝑌 − ( 𝜆 ) , or as 𝜕𝑌 + ( 𝜆 ) 𝑌 − ( 𝜆 ) − 𝜕𝑌 − ( 𝜆 ) 𝑌 − − ( 𝜆 ) = 𝑌 − ( 𝜆 ) 𝜕𝐺 ( 𝜆 ) 𝑌 − ( 𝜆 ) . (31)The Sokhotski-Plemelj formula would then imply (cf. Lemma 2.1 of [2]) thatΘ( 𝜆 ) = ∫︁ Γ 𝑌 − ( 𝑦 ) 𝜕𝐺 ( 𝑦 ) 𝑌 − ( 𝑦 ) 𝑦 − 𝜆 𝑑𝑦 𝜋𝑖 . (32)Substituting 𝑌 ( 𝜆 ) = Ψ( 𝜆 ) 𝑒 ( 𝑖𝑥 𝜆 + 𝑖𝜆 ) 𝜎 in (21), we have 𝑌 ′ ( 𝜆 ) = 𝐴 ( 𝜆 ) 𝑌 ( 𝜆 ) + (︁ 𝑖𝑥 − 𝑖𝜆 )︁ 𝑌 ( 𝜆 ) 𝜎 , (33)and 𝜔 𝑀𝐵 [ 𝜕 ] = ∫︁ Γ Tr( 𝑌 − − 𝑌 ′− ( 𝜕𝐺 ) 𝐺 − ) 𝑑𝜆 𝜋𝑖 = ∫︁ Γ (︁ 𝑖𝑥 − 𝑖𝜆 )︁ Tr( 𝜎 ( 𝜕𝐺 ) 𝐺 − ) 𝑑𝜆 𝜋𝑖 + ∫︁ Γ Tr( 𝐴𝑌 − ( 𝜕𝐺 ) 𝑌 − ) 𝑑𝜆 𝜋𝑖 . (34)We introduce notation 𝑌 ( 𝜆 ) = 𝑃 ( 𝐼 + ∘ 𝑚 𝜆 + 𝑂 ( 𝜆 )) , 𝜆 → , (35) 𝑌 ( 𝜆 ) = (︁ 𝐼 + 𝑚 ( ∞ )1 𝜆 + 𝑂 (︁ 𝜆 )︁)︁ , 𝜆 → ∞ . (36)Substituting these expressions for 𝑌 ( 𝜆 ) to the definition of Θ( 𝜆 ) we getΘ( 𝜆 ) = ( 𝜕𝑃 ) 𝑃 − + 𝑃 ( 𝜕 ∘ 𝑚 ) 𝑃 − 𝜆 + 𝑂 ( 𝜆 ) , 𝜆 → , Θ( 𝜆 ) = 𝜕𝑚 ( ∞ )1 𝜆 + 𝑂 (︁ 𝜆 )︁ , 𝜆 → ∞ . Comparing these formulae with (32) we arrive at the relations ∫︁ Γ 𝑌 − ( 𝑦 ) 𝜕𝐺 ( 𝑦 ) 𝑌 − ( 𝑦 ) 𝑑𝑦 𝜋𝑖 = − 𝜕𝑚 ( ∞ )1 , (37)11 Γ 𝑌 − ( 𝑦 ) 𝜕𝐺 ( 𝑦 ) 𝑌 − ( 𝑦 ) 𝑦 𝑑𝑦 𝜋𝑖 = ( 𝜕𝑃 ) 𝑃 − , (38) ∫︁ Γ 𝑌 − ( 𝑦 ) 𝜕𝐺 ( 𝑦 ) 𝑌 − ( 𝑦 ) 𝑦 𝑑𝑦 𝜋𝑖 = 𝑃 ( 𝜕 ∘ 𝑚 ) 𝑃 − . (39)Let us look now at the last integral in equation (34). Putting in it formula (22) for 𝐴 ( 𝜆 ), wewill see that this integral can be re-written as ∫︁ Γ Tr (︁ 𝐴𝑌 − ( 𝜕𝐺 ) 𝑌 − )︁ 𝑑𝜆 𝜋𝑖 = − 𝑖𝑥 ∫︁ Γ Tr (︁ 𝜎 𝑌 − ( 𝜕𝐺 ) 𝑌 − )︁ 𝑑𝜆 𝜋𝑖 − 𝑖𝑥𝑢 𝑥 ∫︁ Γ Tr (︂ 𝜎 𝑌 − ( 𝜕𝐺 ) 𝑌 − 𝜆 )︂ 𝑑𝜆 𝜋𝑖 + ∫︁ Γ Tr (︂ 𝑃 ( 𝑖𝜎 ) 𝑃 − 𝑌 − ( 𝜕𝐺 ) 𝑌 − 𝜆 )︂ 𝑑𝜆 𝜋𝑖 = − 𝑖𝑥
16 Tr (︁ 𝜎 ∫︁ Γ 𝑌 − ( 𝜕𝐺 ) 𝑌 − 𝑑𝜆 𝜋𝑖 )︁ − 𝑖𝑥𝑢 𝑥 ⎛⎝ 𝜎 ∫︁ Γ 𝑌 − ( 𝜕𝐺 ) 𝑌 − 𝜆 𝑑𝜆 𝜋𝑖 ⎞⎠ + Tr ⎛⎝ 𝑃 ( 𝑖𝜎 ) 𝑃 − ∫︁ Γ 𝑌 − ( 𝜕𝐺 ) 𝑌 − 𝜆 𝑑𝜆 𝜋𝑖 ⎞⎠ . The last equation, with the help of (37) - (39), is transformed into the localized formula, ∫︁ Γ Tr (︁ 𝐴𝑌 − ( 𝜕𝐺 ) 𝑌 − )︁ 𝑑𝜆 𝜋𝑖 = 𝑖𝑥
16 Tr (︁ 𝜎 𝜕𝑚 ( ∞ )1 )︁ − 𝑖𝑥𝑢 𝑥 (︀ 𝜎 ( 𝜕𝑃 ) 𝑃 − )︀ + 𝑖 Tr (︁ 𝜎 𝜕 ∘ 𝑚 )︁ . (40)Substituting the derivative of 𝐺 with respect to 𝜆 in the formula for 𝜃 , we have that 𝜃 [ 𝜕 ] = 12 ∫︁ Γ (︁ 𝑖𝑥 − 𝑖𝜆 )︁ Tr (︁ 𝜎 ( 𝐺 − 𝜕𝐺 − ( 𝜕𝐺 ) 𝐺 − ) )︁ 𝑑𝜆 𝜋𝑖 . Together with (40) this gives us the following formula for 𝜔𝜔 [ 𝜕 ] = 12 ∫︁ Γ (︂ 𝑖𝑥 − 𝑖𝜆 )︂ Tr (︁ 𝜎 (( 𝜕𝐺 ) 𝐺 − + 𝐺 − 𝜕𝐺 ) )︁ 𝑑𝜆 𝜋𝑖 + 𝑖𝑥
16 Tr (︁ 𝜎 𝜕𝑚 ( ∞ )1 )︁ − 𝑖𝑥𝑢 𝑥 (︀ 𝜎 ( 𝜕𝑃 ) 𝑃 − )︀ + 𝑖 Tr (︁ 𝜎 𝜕 ∘ 𝑚 )︁ . One can check directly that Tr (︁ 𝜎 (( 𝜕𝐺 ) 𝐺 − + 𝐺 − ( 𝜕𝐺 )) )︁ ≡ . 𝜔 [ 𝜕 ] = 𝑖𝑥
16 Tr (︁ 𝜎 𝜕𝑚 ( ∞ )1 )︁ − 𝑖𝑥𝑢 𝑥 (︀ 𝜎 ( 𝜕𝑃 ) 𝑃 − )︀ + 𝑖 Tr (︁ 𝜎 𝜕 ∘ 𝑚 )︁ . (41)Substituting (35), (36) to the equation (33), one can express the coefficients of asymptotics of 𝑌 ( 𝜆 ) at 𝜆 = 0 and at 𝜆 = ∞ in terms of 𝑢 . In particular, one gets (cf. [17]), 𝑚 ( ∞ )1 = − 𝑖𝑥𝑢 𝑥 𝑥 𝜎 − 𝑖 (︂ cos 𝑢 − − 𝑢 𝑥 )︂ 𝜎 , ∘ 𝑚 = 𝑖𝑥𝑢 𝑥 𝜎 − 𝑖 (︂ 𝑥
16 (cos 𝑢 − − 𝑥 𝑢 𝑥 )︂ 𝜎 . Inserting these equations together with formula (19) in (41) and using also the fact, that 𝑢 ( 𝑥 )satisfies (1) we transform equation (41) into the final expression for the form 𝜔 in terms of 𝑢 and its derivatives with respect to 𝑥 , 𝑝 and 𝑞 . Proposition 1.
The modified Malgrange-Bertola differential form 𝜔 admits the following rep-resentation 𝜔 = (︂ − 𝑥𝑢 𝑥 𝑥 𝑢 − )︂ 𝑑𝑥 − (︂ 𝑥 𝑢 𝑝 sin 𝑢 + 𝑥 𝑢 𝑥 𝑢 𝑝𝑥 + 𝑥𝑢 𝑥 𝑢 𝑝 )︂ 𝑑𝑝 − (︂ 𝑥 𝑢 𝑞 sin 𝑢 + 𝑥 𝑢 𝑥 𝑢 𝑞𝑥 + 𝑥𝑢 𝑥 𝑢 𝑞 )︂ 𝑑𝑞. (42)We notice that from (42) we have again the statement of Lemma 1, that is that 𝜔 [ 𝜕 𝑥 ] = 𝜕 𝑥 ln 𝜏 − 𝑥 . We want to mention again that this part of the localization formulae has alreadybeen obtained in [2]. We also want to emphasize the important role which is played by the 𝜆 - equation (21) in the derivation of the 𝑝, 𝑞 - part of equation (42). It is the use of thisequation that allowed us to present the original Malgrange-Bertola integral (26), first in theform (34), and then in the localized form (40). In fact, similar technique has already been usedin the study of Toeplitz determinants with the Fisher-Hartwig singularities in paper [3] - seeAppendix 6 and Lemma 6.2 of that paper. Remark 3.
As it was pointed out to the authors by M. Bertola, equation (31) can be usedin the derivations of this section one more time and help to make a significant short cut fromequation (34) to the localized form (40). Indeed, Bertola’s suggestion is to use relation (31) forthe product 𝑌 − ( 𝜆 ) 𝐺 ( 𝜆 ) 𝑌 − ( 𝜆 ) in the last integral of (34) directly and rewrite this integral as ∫︁ Γ Tr (︁ 𝐴 ( 𝜆 ) 𝑌 − ( 𝜆 ) 𝜕𝐺 ( 𝜆 ) 𝑌 − ( 𝜆 ) )︁ 𝑑𝜆 𝜋𝑖 = ∫︁ Γ Tr (︁ 𝐴 ( 𝜆 ) (︁ 𝜕𝑌 + ( 𝜆 ) 𝑌 − ( 𝜆 ) − 𝜕𝑌 − ( 𝜆 ) 𝑌 − − ( 𝜆 ) )︁)︁ 𝑑𝜆 𝜋𝑖 = ∑︁ 𝑝𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 ( 𝜆 ) 𝑑𝜆 𝑟𝑒𝑠 Tr (︁ 𝐴 ( 𝜆 ) 𝜕𝑌 ( 𝜆 ) 𝑌 − ( 𝜆 ) )︁ . (43) This is a quite general construction which allows one to localize the Malgrange-Bertola form foran arbitrary isomonodromic Riemann-Hilbert problem. In our case, one has to evaluate the,properly understood, residues at the points 𝜆 = 0 , ∞ . The result will be equation (40). Proof of Theorem 1
Let us compute 𝑑𝜔 . First we have, 𝑑 [︂(︂ 𝑥𝑢 𝑥 − 𝑥 𝑢 − )︂ 𝑑𝑥 ]︂ = (︂ 𝑥𝑢 𝑥 𝑢 𝑝𝑥 𝑥𝑢 𝑝 sin 𝑢 )︂ 𝑑𝑝 ∧ 𝑑𝑥 + (︂ 𝑥𝑢 𝑥 𝑢 𝑞𝑥 𝑥𝑢 𝑞 sin 𝑢 )︂ 𝑑𝑞 ∧ 𝑑𝑥. Then, using the fact that 𝑢 ( 𝑥 ) satisfies equation (1), we get that 𝑑 [︂(︂ 𝑥 𝑢 𝑝 sin 𝑢 + 𝑥 𝑢 𝑥 𝑢 𝑝𝑥 + 𝑥𝑢 𝑥 𝑢 𝑝 )︂ 𝑑𝑝 ]︂ = (︂ 𝑥𝑢 𝑥 𝑢 𝑝𝑥 𝑥𝑢 𝑝 sin 𝑢 )︂ 𝑑𝑥 ∧ 𝑑𝑝 + (︂ 𝑥 𝑢 𝑝𝑞 sin 𝑢 + 𝑥 𝑢 𝑝 𝑢 𝑞 cos 𝑢 + 𝑥 𝑢 𝑝𝑥 𝑢 𝑞𝑥 + 𝑥 𝑢 𝑥 𝑢 𝑥𝑝𝑞 + 𝑥 𝑢 𝑥 𝑢 𝑝𝑞 + 𝑥 𝑢 𝑝 𝑢 𝑞𝑥 )︂ 𝑑𝑞 ∧ 𝑑𝑝. and 𝑑 [︂(︂ 𝑥 𝑢 𝑞 sin 𝑢 + 𝑥 𝑢 𝑥 𝑢 𝑞𝑥 + 𝑥𝑢 𝑥 𝑢 𝑞 )︂ 𝑑𝑞 ]︂ = (︂ 𝑥𝑢 𝑥 𝑢 𝑞𝑥 𝑥𝑢 𝑞 sin 𝑢 )︂ 𝑑𝑥 ∧ 𝑑𝑞 + (︂ 𝑥 𝑢 𝑝𝑞 sin 𝑢 + 𝑥 𝑢 𝑝 𝑢 𝑞 cos 𝑢 + 𝑥 𝑢 𝑝𝑥 𝑢 𝑞𝑥 + 𝑥 𝑢 𝑥 𝑢 𝑥𝑝𝑞 + 𝑥 𝑢 𝑥 𝑢 𝑝𝑞 + 𝑥 𝑢 𝑞 𝑢 𝑝𝑥 )︂ 𝑑𝑝 ∧ 𝑑𝑞. Adding up the last three equations we obtain that 𝑑𝜔 = 𝑣 𝑝 𝑢 𝑞 − 𝑣 𝑞 𝑢 𝑝 𝑑𝑞 ∧ 𝑑𝑝, where 𝑣 = 𝑥𝑢 𝑥 . From equation (1) it follows that 𝑑𝑑𝑥 ( 𝑣 𝑝 𝑢 𝑞 − 𝑣 𝑞 𝑢 𝑝 ) = 0 , and hence we can observe that 𝑑𝜔 = lim 𝑥 → 𝑑𝜔 = 𝛼 𝑝 𝛽 𝑞 − 𝛼 𝑞 𝛽 𝑝 𝑑𝑞 ∧ 𝑑𝑝 = 𝑑𝛽 ∧ 𝑑𝛼 . Therefore, if we define 𝑤 = 𝜔 + 𝑥 𝑑𝑥 + 𝛼𝑑𝛽 , then the form 𝑤 will be a closed form on the full set of parameters, ( 𝑥, 𝑝, 𝑞 ) and such that 𝑤 [ 𝜕 𝑥 ] = 𝑤 𝐽𝑀𝑈 [ 𝜕 𝑥 ]. This means we can put, 𝜏 = 𝑒 ∫︀ 𝑤 , and this equation would define the tau-function up to a constant, which does not depend on 𝑝 and 𝑞 . Remark 4.
It is worth noticing, that from our analysis it follows that, in the case of thePainlev´e III equation (1), the external differential of the (modified) Malgrange-Bertola form 𝜔 is proportional to the canonical symplectic form (2) for the Hamiltonian dynamics of thePainlev´e equation; indeed, we have that i.e., 𝑑𝜔 = −
14 Ω . 𝑥 asymptotics of the form 𝑤 . Tothis end we shall use asymptotics (4),(5) for 𝑢 ( 𝑥 ) and make the following temporary technicalassumptions, |ℑ 𝜈 | < , |ℑ 𝛼 | < . (44)In our calculations we will need more terms of the large 𝑥 asymptotics at infinity which aregiven in [12], 𝑢 ( 𝑥 ) = 𝑏 + 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − + 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − + 𝑖𝑏 + 𝜈 + 4 𝑖𝜈 − 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − − 𝑖𝑏 − 𝜈 − 𝑖𝜈 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − − 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − − 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − + 𝑂 ( 𝑥 − +5 |ℑ 𝜈 | ) . (45)Substituting this asymptotics at the right hand side of equation (42), we shall arrive, afterrather tedious though straightforward calculations, at the following asymptotic representationof the form 𝜔 as 𝑥 → ∞ , 𝜔 = 𝑑 (2 𝜈𝑥 + 𝜈 ln 𝑥 + 𝜈 ) − 𝑖 𝑏 + 𝑑𝑏 − − 𝑏 − 𝑑𝑏 + ) + (︂ 𝑖𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − − 𝑖𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − )︂ 𝑑𝑥 + 𝑂 ( 𝑥 − |ℑ 𝜈 | ) 𝑑𝑥 + 𝑂 ( 𝑥 − |ℑ 𝜈 | )) 𝑑𝑝 + 𝑂 ( 𝑥 − |ℑ 𝜈 | )) 𝑑𝑞, 𝑥 → ∞ , (46)The derivation of the small 𝑥 asymptotics of the form 𝜔 is based just on the estimate (4), i.e.,no need for its extension, and it is much easy to obtain, 𝜔 = 𝑑 (︂ − 𝛼 𝑥 − 𝛼 )︂ − 𝛼𝑑𝛽 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | )) 𝑑𝑥 + 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ln 𝑥 )) 𝑑𝑝 + 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ln 𝑥 )) 𝑑𝑞, 𝑥 → . (47)As it has already been indicated, the derivations of formulae (46) and (47) are straightforward.However, because of the importance of these formulae for our further analysis, we present thedetails of their derivations in the Appendix.In view of the assumptions (44), estimates (46) and (47) yield the following asymptoticrepresentation for the form 𝑤 , 𝑤 = − 𝑑 (︂ 𝛼 𝑥 + 𝛼 )︂ + 𝑜 (1) , 𝑥 → , (48)and 𝑤 = 𝑑 (︀ 𝜈𝑥 + 𝜈 ln 𝑥 + 𝜈 )︀ − 𝑖 𝑏 + 𝑑𝑏 − − 𝑏 − 𝑑𝑏 + ) + 𝑥 𝑑𝑥 + 𝛼𝑑𝛽 𝑜 (1) , 𝑥 → ∞ . (49)15n the other hand, from (11) and (12) we have that 𝑤 = − 𝑑 (︂ 𝛼 𝑥 )︂ + 𝑑 ln 𝐶 + 𝑜 (1) , 𝑥 → , (50)and 𝑤 = 𝑑 (︂ 𝜈𝑥 + 𝜈 ln 𝑥 + 𝑥 )︂ + 𝑑 ln 𝐶 ∞ + 𝑜 (1) , 𝑥 → ∞ . (51)The comparison of (48) - (49) and (50) - (51) implies that 𝑑 ln 𝐶 = − 𝑑 (︂ 𝛼 )︂ and 𝑑 ln 𝐶 ∞ = 𝑑𝜈 − 𝑖 𝑏 + 𝑑𝑏 − − 𝑏 − 𝑑𝑏 + ) + 𝛼𝑑𝛽 . (52)The last two equations mean that 𝑑 ln 𝐶 ∞ 𝐶 = 𝑑 (︂ 𝜈 + 𝛼 − 𝑖𝜈 )︂ + 𝛼𝑑𝛽 − 𝑖 𝑏 + 𝑑𝑏 − (53)(where we have also taken into account (6)), or thatln 𝐶 ∞ 𝐶 = 𝜈 + 𝛼 − 𝑖𝜈 + 14 ∫︁ ( 𝛼𝑑𝛽 − 𝑖𝑏 + 𝑑𝑏 − ) + 𝑐, (54)where 𝑐 is the numerical constant, independent on 𝑝 and 𝑞 .Following [12], we introduce notation 𝑒 − 𝜋𝑖𝜌 = sin 2 𝜋 ( 𝜎 + 𝜂 )sin 2 𝜋𝜂 . Using this and the connection formulae (7), (10), we can re-write the differential form ( 𝛼𝑑𝛽 − 𝑖𝑏 + 𝑑𝑏 − ) as the differential form in variables 𝜂 , 𝜌 , 𝜎 and 𝜈 ,14 ( 𝛼𝑑𝛽 − 𝑖𝑏 + 𝑑𝑏 − ) = − 𝜋𝑖 ( 𝜎𝑑𝜂 + 𝑖𝜈𝑑𝜌 ) + 2 𝜋𝑖𝑑𝜂 − (12 − 𝜎 ) ln 2 𝑑𝜎 +( 𝑖𝜋 + 4 ln 2) 𝜈𝑑𝜈 + (1 − 𝜎 ) 𝑑 ln Γ(1 − 𝜎 )Γ(2 𝜎 ) + 2 𝑖𝜈𝑑 ln Γ(1 + 𝑖𝜈 ) . Therefore, we can re-write (54) asln 𝐶 ∞ 𝐶 = 𝜈 + 4 𝜎 − 𝜎 − 𝑖𝜈 + 2 𝜋𝑖𝜂 − 𝜎 ln 2 + 24 𝜎 ln 2 + 𝑖𝜋𝜈 𝜈 ln 2 − 𝜋𝑖 ∫︁ ( 𝜎𝑑𝜂 + 𝑖𝜈𝑑𝜌 ) + ∫︁ (1 − 𝜎 ) 𝑑 ln Γ(1 − 𝜎 )Γ(2 𝜎 ) + ∫︁ 𝑖𝜈𝑑 ln Γ(1 + 𝑖𝜈 ) + 𝑐. (55)16t remains to evaluate the integrals in (55). For the integrals involving the Γ-functions one gets, ∫︁ (1 − 𝜎 ) 𝑑 ln Γ(1 − 𝜎 )Γ(2 𝜎 ) = ln Γ(1 − 𝜎 )Γ(2 𝜎 ) − 𝜎 + 8 𝜎 + 2 ln (︁ 𝐺 (1 − 𝜎 ) 𝐺 (1 + 2 𝜎 ) )︁ + 𝑐, (56) ∫︁ 𝑖𝜈𝑑 ln Γ(1 + 𝑖𝜈 ) = 𝑖𝜈 − 𝜈 − 𝑖𝜈 ln(2 𝜋 ) + 2 ln 𝐺 (1 + 𝑖𝜈 ) + 𝑐, (57)where 𝐺 ( 𝑧 ) is the Barnes G-function and we have used the classical formula, ∫︁ ln Γ( 𝑥 ) 𝑑𝑥 = 𝑧 (1 − 𝑧 )2 + 𝑧 𝜋 ) + 𝑧 ln Γ( 𝑧 ) − ln 𝐺 (1 + 𝑧 ) + 𝑐. The most challenging, i.e., the first integral in (55) has already been evaluated in [12]. Here isthe result. ∫︁ 𝜎𝑑𝜂 + 𝑖𝜈𝑑𝜌 = 𝜎𝜂 + 𝑖𝜈𝜌 − 𝒲 ( 𝜎, 𝜈 ) + 𝑐, (58)where the function 𝒲 ( 𝜎, 𝜈 ) is expressed in terms of the dilogarithm 𝐿𝑖 ( 𝑧 ),8 𝜋 𝒲 ( 𝜎, 𝜈 ) = 𝐿𝑖 ( − 𝑒 𝜋𝑖 ( 𝜎 + 𝜂 − 𝑖 𝜈 ) ) + 𝐿𝑖 ( − 𝑒 − 𝜋𝑖 ( 𝜎 + 𝜂 + 𝑖 𝜈 ) ) − 𝜋 𝜂 + 𝜋 𝜈 , (59)Taking into account yet another classical formula, 𝐿𝑖 ( 𝑒 𝜋𝑖𝑧 ) = − 𝜋𝑖 ln ˆ 𝐺 ( 𝑧 ) − 𝜋𝑖𝑧 ln sin( 𝜋𝑧 ) 𝜋 − 𝜋 𝑧 (1 − 𝑧 ) + 𝜋 , where ˆ 𝐺 ( 𝑧 ) = 𝐺 (1 + 𝑧 ) 𝐺 (1 − 𝑧 ) , and the elementary relation,2 cos 𝜋 ( 𝜎 + 𝜂 ± 𝑖𝜈 𝑒 𝑖𝜋 ( ± 𝜎 ∓ 𝜂 − 𝑖𝜈 − 𝜌 ) , (60)we arrive at the following final expression for the first integral in (55) − 𝜋𝑖 ∫︁ 𝜎𝑑𝜂 + 𝑖𝜈𝑑𝜌 = − 𝜋𝑖𝜎𝜂 + 2 ln ˆ 𝐺 ( 𝜎 + 𝜂 + − 𝑖𝜈 )ˆ 𝐺 ( 𝜎 + 𝜂 + 𝑖𝜈 ) − 𝜋𝜂 − 𝑖𝜋𝜈 + 2 𝑖 ln(2 𝜋 ) 𝜈 − 𝜋𝑖𝜈 − 𝜋𝜎 + 6 𝜋𝑖𝜂 + 4 𝜋𝑖𝜎𝜂 − 𝜋𝑖𝜎 + 2 𝜋𝑖𝜂. (61)Substituting formulae (56), (57), and (61) in (55) we arrive at the equation, 𝐶 ∞ 𝐶 = 𝑐 (2 𝜋 ) 𝑖𝜈 𝜈 + 𝜎 − 𝜎 𝑒 𝜋𝑖 ( 𝜂 − 𝜎𝜂 − 𝜎 +2 𝜂 − 𝜎 ) Γ(1 − 𝜎 )Γ(2 𝜎 ) (︃ 𝐺 (1 + 𝑖𝜈 ) 𝐺 (1 + 2 𝜎 ) 𝐺 (1 − 𝜎 ) ˆ 𝐺 ( 𝜎 + 𝜂 + − 𝑖𝜈 )ˆ 𝐺 ( 𝜎 + 𝜂 + 𝑖𝜈 ) )︃ , (62)where 𝑐 is a numerical constant. We know, that if 𝑢 = 0 , 𝜎 = 𝜂 = , 𝜈 = 0, then 𝜏 = const · 𝑒 𝑥 and 𝐶 ∞ = 𝐶 . This choice of parameters satisfies conditions (9). Hence, 𝑐 = 2 𝑒 − 𝑖 𝜋 𝜋 ( 𝐺 ( )) . To complete the proof of Theorem 1 we only need now to lift the technical assumption (44).This can be justified by noticing that the both sides of (62) are analytic functions of theRiemann-Hilbert data. (For the left hand side it follows from the general Birkhoff-Grothendieck-Malgrange theory.)
Remark 5.
The variables ( 𝜂, 𝜎 ) and ( − 𝑖𝜌, 𝜈 ) are canonical variables. In fact, one has that[12], Ω = 32 𝜋𝑖𝑑𝜂 ∧ 𝑑𝜎 = 32 𝜋𝑑𝜌 ∧ 𝑑𝜈. The function 𝒲 was introduced in [12] as the generating function of the canonical transforma-tion ( 𝜂, 𝜎 ) → ( − 𝑖𝜌, 𝜈 ) . Indeed, using (59) , (60) and the fact that 𝐿𝑖 ′ ( 𝑧 ) = − 𝑧 − ln( 𝑧 − , one can show that [12], 𝜂 = 𝜕 𝒲 𝜕𝜎 , and 𝑖𝜌 = 𝜕 𝒲 𝜕𝜈 . The last equation is also equivalent to the integral formula (58).
In [12], and in fact earlier in the pioneering works [5], [6], the derivation of the constantterms in the asymptotics of the tau-functions was based on the heuristic assumption (followedfrom the conformal block representation of the tau-functions) that these constants are related tothe generating functions of the relevant canonical transformations between the canonical pairsassociated with different critical points. In the case of equation (1) the points are 0 and ∞ andthe generating function is the function 𝒲 . This is a very important conceptual point, and ouranalysis justifies it in the case of the Painlev´e III equation (1). It is also worth noticing thatthis hamiltonian interpretation of the ratio 𝐶 ∞ /𝐶 is already present in formula (54). Indeed,this formula tell us that the logarithm of the ratio 𝐶 ∞ /𝐶 is, up to the elementary function, 𝜈 + 𝛼 / − 𝑖𝜈 , the generating function of the canonical transformation between the Cauchydata ( 𝛼, 𝛽 ) and asymptotic at infinity data ( 𝑏 + , 𝑏 − ). In [12] different 𝜏 -function was introduced 𝜏 𝑚 (2 − 𝑥 ) = ( 𝜏 ( 𝑥 )) 𝑥 𝑒 𝑖𝑢 ( 𝑥 )4 , (63)18 𝑚 (2 − 𝑥 ) = 2 − 𝜎 𝑥 𝜎 𝐺 (1 + 2 𝜎 ) 𝐺 (1 − 𝜎 ) (1 + 𝑜 (1)) , 𝑥 → , (64) 𝜏 𝑚 (2 − 𝑥 ) = 𝜒 ( 𝜎, 𝜈 ) 𝑒 𝑖𝜋𝜈 𝜈 (2 𝜋 ) − 𝑖𝜈 𝐺 (1 + 𝑖𝜈 ) 𝑥 𝜈 + 𝑒 𝑥 + 𝜈𝑥 (1 + 𝑜 (1)) , 𝑥 → ∞ . (65)But from formulae (11), (12), (63) we also have 𝜏 𝑚 (2 − 𝑥 ) = 𝐶 𝑒 − 𝑖𝜋 + 𝑖𝜋𝜂 − 𝜎 𝑥 𝜎 Γ(1 − 𝜎 )Γ(2 𝜎 ) (1 + 𝑜 (1)) , 𝑥 → , (66) 𝜏 𝑚 (2 − 𝑥 ) = 𝐶 ∞ 𝑥 𝜈 + 𝑒 𝑥 + 𝜈𝑥 (1 + 𝑜 (1)) , 𝑥 → ∞ . (67)From formulae (64), (65), (66), (67) we get 𝜒 ( 𝜎, 𝜈, 𝜂 ) = 𝐶 ∞ 𝐶 (2 𝜋 ) 𝑖𝜈 − − 𝜈 − 𝜎 +6 𝜎 𝑒 − 𝑖𝜋𝜈 − 𝑖𝜋𝜂 + 𝑖𝜋 𝐺 (1 + 𝑖𝜈 ) 𝐺 (1 + 2 𝜎 ) 𝐺 (1 − 𝜎 ) (︂ Γ(2 𝜎 )Γ(1 − 𝜎 ) )︂ . Substituting here expression for 𝐶 ∞ 𝐶 , we get the formula conjectured in [12] 𝜒 ( 𝜎, 𝜈, 𝜂 ) = (2 𝜋 ) 𝑖𝜈 − 𝑒 𝑖𝜋 ( 𝜂 − 𝜎𝜂 − 𝜎 + 𝜂 − 𝜎 − 𝜈 + ) − 𝐺 ( ) ˆ 𝐺 ( 𝜎 + 𝜂 + − 𝑖𝜈 )ˆ 𝐺 ( 𝜎 + 𝜂 + 𝑖𝜈 ) . One can rewrite (42) 𝜔 = − 𝑥𝑑𝑥 − ℋ 𝑑𝑥 − 𝑥 ℋ 𝑝 𝑑𝑝 − 𝑣𝑢 𝑝 𝑑𝑝 − 𝑥 ℋ 𝑞 𝑑𝑞 − 𝑣𝑢 𝑞 𝑑𝑞 − 𝑥𝑑𝑥 − 𝑑 ( 𝑥 ℋ )4 + 𝑥 ℋ 𝑥 𝑑𝑥 − 𝑣𝑑𝑢 𝑣𝑢 𝑥 𝑑𝑥 . It follows from (1) that 𝑥 ℋ 𝑥 + 𝑣𝑢 𝑥 = ℋ . Using this formula we get 𝜔 = − 𝑑 (︂ 𝑥 𝑥 ℋ )︂ + 14 (︁ ℋ 𝑑𝑥 − 𝑣𝑑𝑢 )︁ . (68)We want to emphasize that all the objects are considered as the functions of the triple ( 𝑥, 𝑝, 𝑞 )and all the differentials are taken with respect to all these three variables.From (68) it follows that up to the multiplication by -4 and the subtraction of a totaldifferential, the Malgrange-Bertola form 𝜔 coincides with the form 𝑣𝑑𝑢 − ℋ 𝑑𝑥 ≡ 𝑣𝑢 𝑥 𝑑𝑥 + 𝑣𝑢 𝑝 𝑑𝑝 + 𝑣𝑢 𝑞 𝑑𝑞 − ℋ 𝑑𝑥. (69)19he restriction of this form on a trajectory of the Hamiltonian system (1), i.e. on the curve, 𝑝 = const , 𝑞 = const , (70)in the extended space of the monodromy data { ( 𝑥, 𝑝, 𝑞 ) } coincides with the form 𝑑𝑆 ( 𝑥 ) = 𝑣𝑢 𝑥 𝑑𝑥 − ℋ 𝑑𝑥, where 𝑆 ( 𝑥 ) is the classical action evaluated on the trajectory (70). Hence, the Malgrange-Bertola form 𝜔 can be treated as a natural extension of the canonical form ℋ 𝑑𝑥 − 𝑣𝑢 𝑥 𝑑𝑥 .It follows then, that the tau-function itself can be identified with the classical action. Moreprecisely, along any classical trajectory, we have that 𝑑 ln 𝜏 ≡ 𝑑 ln 𝜏𝑑𝑥 𝑑𝑥 = − 𝑑𝑆 − 𝑑 ( 𝑥 ℋ ) ≡ (︂ − 𝑑𝑆𝑑𝑥 − 𝑑 ( 𝑥 ℋ ) 𝑑𝑥 )︂ 𝑑𝑥. (71)Of course, this differential identity can be easily (after it is written) checked directly. In itsturn, it allows us to write the following representation for the ratio 𝐶 ∞ /𝐶 in terms of theregularized action integral,ln 𝐶 ∞ 𝐶 = lim 𝑡 → lim 𝑡 → + ∞ ⎛⎝ 𝑡 ∫︁ 𝑡 ℋ − 𝑣𝑢 𝑥 𝑑𝑥 − 𝑥 ℋ ⃒⃒⃒ 𝑡 𝑡 − 𝑡 − 𝜈𝑡 − 𝜈 ln 𝑡 − 𝛼 𝑡 ⎞⎠ . (72)It is worth noticing that, unlike the integral ∫︀ ℋ 𝑑𝑥 , the action integral suits well to the differ-entiation with respect to 𝑝 and 𝑞 ; indeed, after the relevant integration by part the remainingintegral term would disappear in view of (1). Therefore, equation (72) provides us with thepossibility of an alternative derivation of our key formula (53). This derivation would be verysimilar to the evaluation of the action integral of the McCoy-Tracy-Wu solution of the PIIIequation in [13]. Remark 6.
Observe that the extended (with respect to 𝑥 , 𝑝 , 𝑞 ) differential of the form 𝑣𝑑𝑢 − 𝐻𝑑𝑥 is the symplectic form Ω . Therefore, the fact that the Malgrange-Bertola form differs from theform − ( 𝑣𝑑𝑢 − 𝐻𝑑𝑥 ) by a total differential is a fact of general theory; indeed, the (extended)differentials of the both forms coincide – they both are the same 2-form, i.e. − Ω . The additionalinformation we are obtaining in (68) is the explicit evaluation of this total differential. Thisallows us to relate the tau-function and the action differential explicitly which would be importantfor the alterantive evaluation of the tau-constant via the action integral. It might seem quite surprising that one needed to start with the Malgrange-Bertola formin order to discover a rather simple differential identity (71). The absence of the very ideathat the logarithm of the tau-function might differ from the classical action just by a totaldifferential partially explains this. We now believe that the similar fact should be true for anyisomonodrtomy tau-function, although it has been apparently missing in the general monodrmytheory of linear systems . In 2000, the first co-author together with Percy Deift tried to use technique of [13] for evolution of theconstant factors in the asymptotics of the Painlev´e V tau-function associated with the sine-kernel. We failedthen because we did not have the analog of the relation (71) for the Painlev´ve V tau - function we were workingwith. Perhaps, it would make sense to revisit the issue now (although, the relevant constant factors have alreadybeen evaluated since then). cknowledgements The work was supported in part by NSF grant DMS-1361856, and by the SPbGU grant N11.38.215.2014. The authors are also grateful to M. Bertola for very useful comments.
Substitution of the extended asymptotics (45) into the right hand side of (42) leads to thefollowing expressions for its individual terms. ∙ 𝑥𝑢 𝑥 − 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 −
32 (6 𝑖𝜈 + 2 𝜈 + 3 𝑖 )+ 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 −
32 (6 𝑖𝜈 − 𝜈 +3 𝑖 ) − 𝜈 − 𝜈 𝑥 + 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 −
64 + 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 −
64 + 𝑂 ( 𝑥 − |ℑ 𝜈 | ); ∙ − 𝑥 𝑢 −
1) = 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝜈 − 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − − 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝑏 𝑒 𝑖𝑥 𝑥 𝑖𝜈 −
32 (6 𝑖𝜈 + 2 𝜈 − 𝑖 ) − 𝑏 − 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 −
32 (6 𝑖𝜈 − 𝜈 − 𝑖 ) + 𝑂 ( 𝑥 − |ℑ 𝜈 | ); ∙ 𝑥 𝑢 𝑝 sin 𝑢 = 𝑏 + 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈
16 (6 𝑖𝜈 + 5 𝜈 − 𝑖 ) + 𝑏 − 𝑏 + 𝑝 𝑥 𝑏 + 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 +1 − 𝑏 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − 𝑏 − 𝑏 + 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈
64 + 𝑏 − 𝑏 + 𝑝 𝜈 𝑖𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 +1 ln 𝑥 − 𝑖𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 +1 ln 𝑥 − 𝑖𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 ln 𝑥 − 𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 ln 𝑥
16 (6 𝜈 +2 𝑖𝜈 − − 𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 ln 𝑥
16 (6 𝜈 − 𝑖𝜈 − 𝑏 − 𝑝 𝑏 + 𝑥 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 +1 − 𝑏 𝑏 − 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈
16 + 𝑏 + 𝑏 − 𝑝 𝜈 𝑖𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 ln 𝑥 − 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈
16 (6 𝑖𝜈 − 𝜈 − 𝑖 ) + 𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 𝑖𝜈 − − 𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 𝑖𝜈 + 1) + 𝜈 𝑝 𝜈 + 𝑂 ( 𝑥 − |ℑ 𝜈 | ); ∙ 𝑥 𝑢 𝑞 sin 𝑢 = {︁ 𝑝 → 𝑞 }︁ ; 21 𝑥 𝑢 𝑥 𝑢 𝑝𝑥 = − 𝑏 + 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 +1 − 𝑖𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 +1 ln 𝑥 − 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 +1 𝑖𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 +1 ln 𝑥 − 𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 𝑖𝜈 +1) − 𝑏 + 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈
16 (6 𝑖𝜈 + 𝜈 +3 𝑖 )+ 𝑏 − 𝑏 + 𝑝 𝑥 𝑏 + 𝑏 − 𝑝 𝑥 − 𝜈 𝑝 𝜈 + 𝑏 + 𝑏 − 𝑝 𝜈 𝑏 − 𝑏 + 𝑝 𝜈 𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 ln 𝑥
16 (6 𝜈 − 𝑖𝜈 + 3) + 𝑏 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 𝑖𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 ln 𝑥
16 + 𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 ln 𝑥
16 (6 𝜈 +2 𝑖𝜈 +3)+ 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈
16 (6 𝑖𝜈 − 𝜈 +3 𝑖 )+ 𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 𝑖𝜈 −
1) + 𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝑖𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 ln 𝑥 − 𝑏 − 𝑏 + 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝑏 𝑏 − 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈
64 + 𝑂 ( 𝑥 − |ℑ 𝜈 | ); ∙ 𝑥 𝑢 𝑥 𝑢 𝑞𝑥 = {︁ 𝑝 → 𝑞 }︁ ; ∙ 𝑥𝑢 𝑥 𝑢 𝑝 𝑖𝑏 + 𝑏 + 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 − 𝑏 𝜈 𝑝 𝑒 𝑖𝑥 𝑥 𝑖𝜈 ln 𝑥 𝑖𝑏 + 𝑏 − 𝑝 − 𝜈 𝑝 𝜈 ln 𝑥 − 𝑖𝑏 − 𝑏 + 𝑝 − 𝑖𝑏 − 𝑏 − 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 − 𝑏 − 𝜈 𝑝 𝑒 − 𝑖𝑥 𝑥 − 𝑖𝜈 ln 𝑥 𝑂 ( 𝑥 − |ℑ 𝜈 | ); ∙ 𝑥𝑢 𝑥 𝑢 𝑞 {︁ 𝑝 → 𝑞 }︁ . It is quite remarkable that the substitution of these long expressions into the right hand sideof (42) yields to the compact formula (46).For asymptotics at zero we get the following estimates. 𝑥𝑢 𝑥 − 𝑥 𝑢 −
1) = 𝛼 𝑥 + 𝑂 ( 𝑥 −|ℑ 𝛼 | ) ,𝑥 𝑢 𝑝 sin 𝑢 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ) , 𝑥 𝑢 𝑞 sin 𝑢 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ) ,𝑥 𝑢 𝑥 𝑢 𝑝𝑥 𝑥 = 𝛼𝛼 𝑝 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ) , 𝑥 𝑢 𝑥 𝑢 𝑞𝑥 𝑥 = 𝛼𝛼 𝑞 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ) ,𝑥𝑢 𝑥 𝑢 𝑝 𝛼𝛼 𝑝 ln 𝑥 𝛼𝛽 𝑝 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ln 𝑥 ) ,𝑥𝑢 𝑥 𝑢 𝑞 𝛼𝛼 𝑞 ln 𝑥 𝛼𝛽 𝑞 𝑂 ( 𝑥 −|ℑ ( 𝛼 ) | ln 𝑥 ) . These equations yield at once (47).
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