Interpolating function and Stokes Phenomena

Transcription

1 Interpoating function and Stokes Phenomena Masazumi Honda and Dieep P. Jatkar arxiv: v3 [hep-th] 2 Ju 205 Harish-Chandra Research Institute Chhatnag Road, Jhunsi Aahabad 209, India Abstract When we have two expansions of physica quantity around two different points in parameter space, we can usuay construct a famiy of functions, which interpoates the both expansions. In this paper we study anaytic structures of such interpoating functions and discuss their physica impications. We propose that the anaytic structures of the interpoating functions provide information on anaytic property and Stokes phenomena of the physica quantity, which we approximate by the interpoating functions. We expicity check our proposa for partition functions of zero-dimensiona ϕ 4 theory and Sine-Gordon mode. In the zero dimensiona Sine-Gordon mode, we compare our resut with a recent resut from resurgence anaysis. We aso comment on construction of interpoating function in Bore pane. HRI/ST/50 masazumihonda@hri.res.in dieep@hri.res.in

2 Contents Introduction 2 Interpoating function 4 3 Partition function of zero-dimensiona ϕ 4 theory 5 3. Interpoation aong positive rea axis Interpoation aong negative rea axis Partition function of zero-dimensiona Sine-Gordon mode 4 4. Interpoation aong positive rea axis Interpoation aong negative rea axis Concusion and Discussions 8 A Comparison with resurgence resut 2 B Comments on interpoation in Bore pane 22 B. Method B.2 Comparison with usua interpoating function in 0d Sine-Gordon mode C Expicit forms of interpoating functions 24 C. Partition function of 0d ϕ 4 theory C.. Interpoation aong positive rea axis of g C..2 Interpoation aong negative rea axis of g C.2 Partition function of 0d Sine-Gordon mode C.2. Interpoation aong positive rea axis of g C.2.2 Interpoation aong negative rea axis of g C.2.3 Bore-FPR References 26 Introduction In most non-exacty sovabe probems, one tries to ook for a sma parameter or a arge parameter so that we can set up a perturbative expansion. This perturbation series is typicay an asymptotic series which is divergent and has zero radius of convergence. Often the perturbation series takes different forms depending on the argument of its expansion parameter. This behavior is known as the Stokes phenomenon []. For exampe, if we consider a physica quantity having an integra representation as in quantum fied theory and string theory, then the integrand usuay has mutipe sadde points and its perturbative expansion with the parameter g typicay takes the form a 0,k g k +e S g) a,k g k +e S 2g) a 2,k g k +,.) k k k

3 where S i g) is the action evauated at each sadde point and each sum denotes the sma-g expansionaroundthatsaddepoint. Instandardsituations, theweighte S ig) isexponentiay suppressed for rea positive g and vanishes as approaching g +0. In other words a the terms except the first one in.) describe non-perturbative effects. However, if we change argg), then S i g) might have negative rea part. For this case, the weight e S ig) is not exponentiay suppressed but exponentiay growing. This type of transition happens when we cross an anti-stokes ine given by ReS i ) = 0. On the other hand, the coefficients a i,k themseves mightjumpaswechangeargg). ThishappenswhenwecrossaStokesinedefined by ImS i ) = ImS j ). Thus Stokes phenomena has deep connections to non-perturbative effects. In most practica situations, we can access to ony first few terms of a perturbation series around singe sadde point for particuar argument of expansion parameter. For this case, it is hard to find when Stokes phenomena occur and when contributions from other sadde points become important. In this paper we deveop a too to study Stokes phenomena in somewhat specia situations. Sometimes we can have two perturbative expansions of physica quantity around two different points in parameter space, for exampe, in theory with S-duaity, fied theory with gravity dua, attice gauge theory with weak and strong couping expansions, statistica system with high and ow temperature expansions, and so on. One can then construct functions, which interpoate these two expansions. The most standard approach to this is two-point) Padé approximant, which is a rationa function having the two expansions up to some orders. Recenty Sen has considered another type of interpoating function, which has the form of a Fractiona Power of Poynomia FPP) [2]. After a whie, one of us has constructed a more genera cass of interpoating functions described by Fractiona Powers of Rationa function FPR) [3], which incudes the Padé approximant and FPP as specia cases. It has turned out that these interpoating functions usuay provide better approximations than each perturbative expansion in intermediate regime of the parameter, see [2, 3, 4, 5, 6, 7] for various appications. Athough these are quite nice as first attempts, properties of interpoating functions themseves have not been extensivey studied yet. In this paper we propose new properties of the interpoating functions. We focus on anaytic structure of the interpoating functions 2 by treating them as compex functions and discuss their physica impications. The FPR is some power of a rationa function and the rationafunctionhas poesandzeros. When thepower isnot integer, then thepoes andzeros of the rationa function give rise to branch cuts of the FPR. Here we propose that the branch cuts of the FPR encode information about the anaytic property and Stokes phenomena of the physica quantity, which we try to approximate. More concretey we propose that each branch cut of the FPR has the foowing possibe interpretations. There is aso another type of interpoating functions [8], which is not specia case of the FPR. This has been appied to ON) non-inear sigma mode. 2 Note that our interpoating function approach is different from resurgence approach, which has been recenty studied in a series of works[9, 0,, 2, 3, 4, 5, 6, 7, 8, 9, 20, 2, 22, 23]. Whie we use singe types of weak and strong couping expansions as input data, the resurgence approach uses weak couping expansions around mutipe sadde points and does not use strong couping expansion. 2

4 . The branch cut is particuar to the FPR and the artifact of the approximation. Namey, this type of branch cut is not usefu to extract physica information. 2. The physica quantity, which we approximate by the FPR, has a branch cut near from the branch cut of the FPR. Namey, the branch cut of the FPR we approximates the true branch cut of the physica quantity. 3. Near the branch cut, one of perturbation series of the physica quantity changes its dominant part. This case is further separated into the foowing two possibiities. a) We have an anti-stokes ine of the perturbative expansion near the branch cut. Namey, athough the perturbative series itsef does not change its own form, contributions from other sadde points become dominant across the ine. This possibiity ikey occurs for first branch cut measured from a specific axis where we construct interpoating functions. b) The perturbative series itsef does change the form. Namey, we have a Stokes ine near the branch cut, whose diagona mutipier is different from. When we have a Stokes ine across which sub-dominant parts of the perturbative series change, the FPR cannot detect this type of Stokes ine. We expicity check our proposa in two exampes: partition functions of the ϕ 4 theory and the Sine-Gordon mode in zero dimensions. Simiar features seem to appear aso in other exampes such as BPS Wison oop in 4d N = 4 Super Yang-Mis theory, energy spectrum in d anharmonic osciator etc [24]. We expect that our resut is appicabe in more practica probems, where we do not know exact soutions. One possibe utiity of such an anaysis is that we can anticipate anaytic property and Stokes phenomena of physica quantity by ooking at anaytic structures of interpoating functions. This paper is organized as foows. In section 2 we introduce our interpoating functions described by the fractiona power of rationa functions FPRs). In section 3 we study interpoating probem in the 0d ϕ 4 theory in great detai. We ook at the anaytic property of the interpoating function as a compex function and propose its physica interpretation. In section 4 we anayze the 0d Sine-Gordon mode and check that our proposa is true aso for this mode. Section 5 is devoted to concusion and discussions. In appendix A we compare our interpoating function with a recent resut from resurgence anaysis [20] in the 0d Sine- Gordon mode. Our resut impies that the FPR and resurgence pay compementary roe with each other. In appendix B we expain an attempt to construct interpoating function in Bore pane and test its utiity in the 0d Sine-Gordon mode. In appendix C we write down expicit forms for interpoating functions used in the main text. 3

5 2 Interpoating function We introduce the interpoating functions in this section, which is essentiay a review of [3]. Suppose that we wish to determine a function Fg), which has 3 the sma-g expansion F s Ns) g) and arge-g expansion F N ) g) taking the forms F Ns) s N s g) = g a k=0 s k g k, F N ) We can then naivey expect that these expansions approximate Fg) as N g) = g b k g k. 2.) Fg) = F Ns) s g)+og a+ns ) = F N ) g)+og b N ). 2.2) Athough this seems to be a somewhat imited case, this situation incudes a arge cass of physica probems, e.g., theory with S-duaity, fied theory with gravity dua, attice gauge theory with weak and strong couping expansions, statistica system with high and ow temperature expansions, and etc. In terms of the two expansions, one can construct the foowing function [3] where p = 2 F α) m,n g) = s 0g a [ + p k= c kg k + q k= d kg k k=0 ] α, 2.3) m+n+ a b ), q = m+n++ a b ). 2.4) α 2 α The coefficients c k and d k are determined such that series expansions of F m,ng) α) around g = 0 and g = reproduce the sma-g and arge-g expansions 2.) of Fg) up to Og a+m+ ) and Og b n ), respectivey. Due to this property, the function F m,ng) α) interpoates the smag and arge-g expansions up to these orders. Since the interpoating function is usuay 4 described by the Fractiona Power of Rationa function, we ca this type of the interpoating function FPR. Note that the rationa function inside the square bracket in 2.3) is a ratio of poynomias, i.e., p,q Z 0, 2.5) which eads the foowing condition α = { a b 2+ for m+n : even a b 2 for m+n : odd, with Z. 2.6) It is now easy to see that the FPR incudes the Padé approximant and the Fractiona Power of Poynomia FPP) constructed in [2] as specia cases. If we take 2+ = a b for a b Z and m+n to be even, then this becomes the Padé approximant whie if we take 2+ = m+n+ 2 = m+n+) for even odd) m+n then we get the FPP. Therefore we beow refer to aso the Padé and FPP as FPR. 3 More generaywe might haveperturbative expansionsaround g = g and g = g 2 with g 2 > g and woud ike to construct their interpoating functions. However, if we change the variabe as x = g g )/g 2 g), then this probem is reduced to interpoating probem of sma-x and arge-x expansions. Thus our setup does not ose generaity in this sense. 4 When a b is irrationa number, the power α is irrationa number. 4

6 Figure : Summary of Stokes phenomena for the sma-g expansion in the partition function of the 0d ϕ 4 theory. The bue soid ines denote the Stokes ines whie the green dashed ines denote the anti-stokes ines. 3 Partition function of zero-dimensiona ϕ 4 theory In this section we study interpoation probem for the partition function of the 0d ϕ 4 theory. Athough this exampe has been aready studied we in [2, 3] for rea non-negative couping constant, here we consider genera compex couping. As mentioned above, we study anaytic properties of interpoating functions and their physica impications. Let us consider the integra Fg) = dx e x2 2g x4, 3.) g which can be exacty performed as Fg) = πe32g 2 4g [ I 4 [ πe32g 2 I 4 g g 2 ) I 4 32g 2 )+I 4 )] 32g 2 )] 32g 2 for Reg) > 0 for Reg) 0, 3.2) where Iz) is the modified Besse function of the first kind. The sma-g expansion of Fg) depends on the argument of g, and its dependence is given by 2π 3 2πg π Fg) = π 2 05g4 6g 2 +2+Og 6 )) +Og 6 ) for argg) π, π) πie 2g g 2 05g 4 +6g 2 +2+Og 6 )) for argg) π,π) 2 π 2 05g4 6g 2 +2+Og 6 ))+. πie 6g 2 05g 4 +6g 2 +2+Og 6 )) for argg) π, π) 2 3.3) This dependence on argg) ceary refects the fact that we have Stokes ines oriented aong argg) = ±π/2 and π, across which the form of the sma-g expansion changes. Notice that across the ray argg) > 3π/4, the terms invoving the exponentia factor in ine 2 and 3 in eq. 3.3)) are dominant compared to those without the exponentia factor. This indicates that we have the anti-stokes ines oriented aong argg) = ±3π/4. It is easy to understand this behavior from the standard sadde point anaysis for g. 5

7 Sadde points x of the integration are given by x = 0,x ±, with x ± = ± i 2 g. 3.4) Then, the action Sx) = x2 2g +x4 at the sadde points x takes the vaues Sx = 0) = 0, Sx = x ± ) = 6g2. 3.5) For argg) < π/2, we can pick up ony the trivia sadde point x = 0 by deforming the origina integra contour, ) to a steepest descent contour, whie we can pick up a the sadde points otherwise. We have reative minus sign in the contributions from the non-trivia saddepointsx = x ± becausedirectionsofthesteepest descent throughx = x ± areopposite between the cases for π/2 < argg) < π and π < argg) < π/2. Note that the rea parts of the action at the non-trivia sadde points change their signs across argg) = ±3π/4. This means that we have anti-stokes ines of the sma-g expansion at argg) = ±3π/4. This discussion is summarized in fig.. The arge-g expansion, on the other hand, is independent 5 of argg): Fg) = g /2 2 Γ/4)+ 6 Γ /4)g + ) 64 Γ/4)g 2 +Og 3 ). 3.6) 3. Interpoation aong positive rea axis Let us first take the couping g to be rea and positive as usua. For this case, we have the foowing sma-g and arge-g expansions F Ns) s g) = N s k=0 F N ) g) = g /2 2Γ2k +/2) s k g k, s 2k+ = 0, s 2k = 4) k, 3.7) k! N =0 k g k, k = Γ k + ) 2 4 k, 3.8) 2k! 2) which are compared with the exact resut in fig. 2 [Left]. In terms of these expansions, we can construct FPR-type interpoating function F m,ng) α) see app. C. for expicit forms). In fig. 2 [Right], we test vaidity of the FPRs by potting F α) m,n Fg), Fg) against g for some m, n, α). We easiy see that these interpoating functions provide good approximations to the origina function Fg). Especiay F /2) 6,6 g) approximates the exact 5 These behaviors of the expansions can be understood aso from viewpoint of differentia equation for the Besse function athough we can know this information after finding the exact resut. The modified Besse function has essentia singuarity at g = 0 and has ony a branch cut singuarity at g =. The Stokes phenomenon of the sma-g expansion is a manifestation of the essentia singuarity. There is no such behavior near the branch point, and the arge g expansion is independent of argg). 6

8 Figure 2: [Left] The partition function of the 0d ϕ 4 theory back soid), its sma-g expansions F s Ns) g) red dotted) and arge-g expansions F N ) g) bue dashed) for argg) = 0. [Right] Comparison of the FPRs with some m,n,α) on the non-negative rea axis of g. resut Fg) very we: the maxima vaue of the ratio F /2) 6,6 g) Fg))/Fg) is O0 8 ). Thus we find that our interpoating scheme in this exampe works quite we at east aong the positive rea axis of g. Of course this resut is not new and has been aready seen in the previous studies [2, 3]. Here we ask another question. Suppose we perform naive anaytic continuation of the interpoating function F α) m,ng) to the whoe compex pane of g. Then, does the interpoating function F /2) 6,6 g), which is very precise aong the positive rea g, sti gives a good approximation beyond the positive rea axis? In order to answer this question, we pot the quantity 6 F /2) 6,6 g)/fg) against g for some argg) in fig. 3. First, we easiy observe in fig. 3 [Left top] that the FPR gives very precise approximation for argg) = π/0, whose reative error is O0 7 ) at worst. Fig. 3 [Right top] shows that this is true aso for argg) = 4π/0, abeit not as accurate as that for argg) = π/0. We therefore concude that the FPR can give good approximation of the exact function even beyond the rea positive axis. However, as we further increase argg) to π, the approximation starts becoming worse. As seen in fig. 3 [Left bottom], the FPR sti gives good approximation for argg) = 6π/0 but the reative error becomes O0.%). Since we have the exponentiay suppressed corrections in the weak couping regime forπ/2 < argg) < 3π/4,whichcomesfromthenontriviasaddepoints, werecognizethatthe exponentiay suppressed corrections are responsibe for this error. For 3π/4 < argg) < 5π/4, the contributions from the nontrivia sadde points become exponentiay growing. Since the FPR acks this information, the FPR shoud show very arge error for 3π/4 < argg) < 5π/4 in sma- g regime. Indeed we have O00%) error on the negative rea axis as seen in fig. 3 [Right bottom]. In fig. 4, we summarize vaidity of approximation by the FPR. The shaded part shows the region where the FPR has more than 5% reative error. We aso draw the zeros and poes of the rationa function F α) 6,6 ) 2 associated 7 with the FPR, which give branch cuts of the FPR. From this figure we observe some points: 6 Note that F /2) 6,6 g) Fg))/Fg) does not take rea vaue for compex g in genera. 7 If we consider genera FPR F m,n, α) then its natura associated rationa function is F α) m,n) / α. 7

9 Figure 3: The quantity F /2) 6,6 g) Fg))/Fg) is potted against g for some argg). The shaded region ooks ike a fan, whose radia bounding ines are cose to the anti- Stokes ines of the sma-g expansion. This is natura since the dominant part of the sma-g expansion changes across the anti-stokes ines and the FPRs do not know this information in sma- g regime. The radius of the fan shoud be finite since the FPR gives the correct arge-g expansion by construction even across the anti-stokes ines. The boundary of the shaded) fan-ike region is simiar to the region surrounded by the origin, poes and zeros of the rationa function F /2) 6,6 g)) 2. Especiay the anti-stokes ines are cose to the ines between the origin and the first poes measured from the positive rea axis 8. This woud be natura because when the exact function Fg) does not have singuarities around the poes of F /2) 6,6 g)) 2, then the FPR differs from the exact function by a arge amount in the neighborhood of its poes. Hence for this case, the FPR ceary gives bad approximation around the poes. There is a branch cut from the poe at the negative rea axis to g =. Since the partition function Fg) has the branch cut on 0, ), we interpret that the branch cut of the FPR approximates the true branch cut of the exact resut. Athough the sma-g expansion has the Stokes ine at the imaginary axis, the FPR does not detect this Stokes ine. This is because the dominant part of the sma-g expansion does not change across this Stokes ine and ony the sub-dominant part changes. 8 The first poes are ocated at argg) ±2.358 whie the anti-stokes ines are oriented aong argg) = 3π/

10 Figure 4: The region where the interpoating function F /2) 6,6 g) gives bad approximation. In the shaded region, the ratio F /2) 6,6 /F is arger than We aso pot zeros the symbo + ) and poes the symbo ) of the rationa function F /2) 6,6 g)) 2 associated with the FPR. The bue soid ines denote the branch cuts of F /2) 6,6 g). These resuts ead us to the conjecture about the genera feature of FPR that each branch cut of the FPR has the foowing possibe interpretations.. The branch cut is an artifact of the approximation by the FPR. 2. The physica quantity Fg) has a branch cut near the branch cut of the FPR. 3. Near the branch cut, one of perturbation series of Fg) changes its dominant part. This case impies the foowing two possibiities on Stokes phenomena. a) We have an anti-stokes ine of the perturbative expansion near the branch cut. This possibiity occurs most ikey for the first branch cut measured from the specific axis where the interpoating function is constructed. b) We have a Stokes ine near the branch cut, whose diagona mutipier is different from. One of immediate questions here is if these features are particuar for this probem or true aso for other probems. We wi expicity check this in sec. 4 that this is true aso for the partition functions of the 0d Sine-Gordon mode. This seems to hod aso in other exampes such as BPS Wison oop in 4d N = 4 Super Yang-Mis theory, energy spectrum in d anharmonic osciator etc [24]. Another important question is if we can construct another interpoating function, which gives good approximation in the region, where the interpoating function aong the positive rea axis becomes bad. One natura way to do this is to construct interpoating functions aong a specific axis with argg) > 3π/4, where dominant part of the 9

11 Figure 5: Simiar pot as fig. 4 for the FPP F /26) 6,6 g). sma-g expansion comes from the non-trivia sadde points, and then to extend these to the compex g. In next subsection, we wi perform this by considering interpoating functions aong the negative rea axis. Remarks One might ask if the FPRs F m,n α) with the same m,n) but different α gave simiar resuts. In this exampe we find that the resuts strongy depend on α. For instance, et us ook at fig. 5, which is simiar to the pot in fig. 4 but for F /26) 6,6 g). Note that the FPR for this case becomes the FPP [2]. On the rea positive axis, this interpoating function gives maximum error of about 0.6%. It is easy to see from fig. 5 that the anaytic structure of F /26) 6,6 g) is very different from the one of F /2) 6,6 g). Especiay, when we start from the rea positive axis and go towards the anti-stokes ine at argg) = 3π/4, we encounter many branch cuts for this case. Another important difference is that the first branch cuts from the rea positive axis are ocated at the pretty smaer ange argg) ±.2, which is quite far from the ocation of the anti-stokes ikes. This seems to be the reason why the FPP F /26) 6,6 g) gives the worse approximation than F /2) 6,6 g). 3.2 Interpoation aong negative rea axis In order to construct another interpoating function precise for argg) > 3π/4, et us consider interpoating function aong the negative rea axis of g and then consider its naive anaytic continuation to the whoe compex pane. The function Fg) has the foowing sma-g and 0

12 Figure 6: [Left] The function Fg) = ie 6g 2 Fg) back soid), its sma-g expansions F s Ns) g) red dotted) and arge-g expansions F N ) g) bue dashed) for argg) = π ǫ. [Right] The quantity F m,nt)/ Ft) ) α) = F α) t)/f t) ) is potted to t = g. L+,m,n arge-g expansions on ǫ-neighborhood of the negative rea axis: F t+iǫ) = πie 6t 2 2+6t 2 +Ot 4 ) ) + π 2 6t 2 +Ot 4 ) ) +Oǫ) 2 = it /2 2 Γ/4) 6 Γ /4)t + ) 64 Γ/4)t 2 +Ot 3 ) +Oǫ), F t iǫ) = + πie 6t 2 2+6t 2 +Ot 4 ) ) π + 2 6t 2 +Ot 4 ) ) +Oǫ), 2 = +it /2 2 Γ/4) 6 Γ /4)t + ) 64 Γ/4)t 2 +Ot 3 ) +Oǫ), 3.9) with t R +. Namey, the dominant parts of the sma-g expansion and the arge-g expansion change their signs across the negative rea axis. This refects that the exact function Fg) has the square root branch cut on the negative rea axis. Instead of Fg), et us consider interpoating functions of the quantity: Ft) = im 6g 2 Fg +iǫ), t R ) g t ǫ +0 ie The function Ft) has the sma-t and arge-t expansions, Ft) = 2 π +6 πt πt πt πt 8 +O t 0) 4 = t /2 π 2Γ 3 ) + πt 8 2Γ πt 2 ) Γ πt 3 ) Γ ) +O t 4)),3.) where we have dropped the exponentiay suppressed correction Oe 6t 2 ) coming from the trivia sadde point in the sma-t expansion. Denoting interpoating function of Ft) as F α) m,nt), we can approximate the origina function by using the function F α) L±,m,n g) = ie+ 6g 2 Fα) m,n g). 3.2)

13 Figure 7: [Left] The region where the ratio F /2) L+,0,0 /F is arger than We aso pot zeros +) and poes ) of the rationa function F /2) 0,0 )2. The bue soid ines denote the branch cuts of F /2) L+,0,0. [Right] Simiar pot as the eft for F/2) L,0,0. The interpoating function F α) L±,m,n g) reproduces the sma-g and arge-g expansions of Fg) for g R ±iǫ up to certain orders. Indeed F α) L+,m,n g) approximates Fg) quite we aong g)has aboutreative 0.8% error at worst. Let us consider genera compex g in the interpoating function F α) L±,m,n g) as in the ast subsection. Then, uness we cross the anti-stokes ine or branch cuts, F α) L±,m,n g) gives the thenegative rea +iǫ) axis 9 asseen in fig. 6[Right]. Especiay F α) L±,0,0 correct arge-g expansion and dominant part of sma-g expansion. If we cross the anti- Stokes ine, then the sma-g expansion is dominated by the contribution from the trivia sadde point and F α) L±,m,n g) shoud fai to approximate Fg) in sma- g regime. Aso, if we cross the branch cut particuar to the FPR, then the FPR wi pick up an extra phase and aso break the approximation. This extra phase coud be trivia depending on the number of times, where we cross branch cuts. The vaidity of the approximation by F /2) L±,0,0, g) is summarized with its anaytic property in fig. 7. One of important differences from the interpoation aong the rea positive axis is that the semi-)circuar branch cut of F /2) L±,0,0 g) surrounds a the poes and zeros on the right pane. Therefore if we go across the circuar branch cut on the right haf pane, then F /2) L±,0,0 g) undergoes a sign fip and hence it fais to approximate Fg) across the circuar branch cut. However, these FPRs aso have a branch cut aong the positive rea axis and crossing this branch cut on the positive rea axis eads to another fip in the sign. As a resut F /2) L±,0,0 g) recovers the correct sign and gives the good approximation to Fg) again. That is why we see the disconnected unshaded regions in fig. 7. Another important difference is that the first branch cuts as measured from the negative 9 Simiar resut hods aso for F α) L,m,n g). 2

14 Figure 8: [Left] First poes and zeros of F /2) m,n ) 2 from the negative rea axis for various m,n). The symbo + denotes zeros and * denotes poes. [Right] The region where the patch F g) of the best FPRs gives more than reative 5% error. rea axis sighty deviate from the anti-stokes ines. Whie the anti-stokes ines are oriented aong argg) = ±3π/4 ±2.356, these first cuts are ocated at the ange 0 argg) = ± However, we expect that as increasing the vaues of m,n) in the FPR F m,n /2), the first poes of F m,n /2) ) 2 wi approach to the first zeros from the negative rea axis and the first cuts wi finay vanish for sufficienty arge m,n). Indeed we can easiy observe in fig. 8 [Left] that the pair of first zeros and poes seem to converge to the same point as increasing m,n). Thus we concude that the first pair of the branch cuts are the artifact of the approximation by the FPR with insufficienty arge m, n). We ca this type of singuarities fake singuarities. The above resut woud be natura because the FPR with arger m,n) tends to give better approximation in this probem and may improve the vaidity of the approximation near the anti-stokes ines. In next section we wi see that FPRs in the 0d Sine-Gordon mode have simiar features. Finay et us find good approximation of Fg) in region as wide as possibe by patching the best interpoating functions aong the positive and negative rea axis. In fig, 8 [Right], 0 Note that when we write vaues of argg), we aways denote the vaues measured with respect to the positive rea axis with countercockwise. When either ofthe sma-g or arge-g expansion is convergentas in this probem, we expect this tendency because the convergent expansion itsef gives very precise approximation inside its radius of convergence and we can regard their FPRs with arge m, n) as anaytic continuation of the convergent expansion to whoe range of vaues of g. However, it is nontrivia in genera whether the decreasing behavior of errors in the FPRs wi be monotonic or not, athough the error of the particuar FPR F m,n /2) in this probem seems to be monotonicay decreasing. For exampe, in 2d Ising mode with finite voume, both ow and high temperature expansions of specific heat are convergent For this case, error of its FPR with fixed α tends to decrease but not monotonicay decrease as increasing m, n) [3]. 3

15 g trivia & exp. growing trivia & exp. decaying trivia & exp. growing trivia & exp. decaying Figure 9: Summary of Stokes phenomena for the sma-g expansion in the partition function of the 0d Sine-Gordon mode. The bue soid ines denote the Stokes ines whie the green dashed ines denote the anti-stokes ines. we draw range of vaidity of approximation by F /2) 6,6 g) for argg) < 3π/4 F g) = F /2) L+,0,0 g) for 3π/4 < argg) < π. 3.3) F /2) L,0,0 g) for π < argg) < 3π/4 This indicates that the patching F g) has 5% or better accuracy in the very wide region. 4 Partition function of zero-dimensiona Sine-Gordon mode Let us consider the partition function of the zero-dimensiona Sine-Gordon mode: Fg) = g π/2 π/2 dx e 2g sin2x, 4.) which was considered by Cherman-Koroteev-Unsa in the context of resurgence [20]. As in the ast section, this integra can be evauated exacty as Fg) = π ) e 4g I0. 4.2) g 4g The function Fg) has the foowing sma-g and arge-g expansions π 8+4g 32 +9g2 +Og 3 )) for argg) = 0 Fg) = π g +9g2 +Og 3 )) i π 32 e 2g 8 4g +9g 2 +Og 3 )) for argg) 0,π) π 8+4g 32 +9g2 +Og 3 ))+i, π 32 e 2g 8 4g +9g 2 +Og 3 )) for argg) π,0) 4.3) 4

16 Figure 0: [Left] The partition function of the 0d Sine-Gordon mode back soid), its sma-g expansions F s Ns) g) red dotted) and arge-g expansions F N ) g) bue dashed) for argg) = 0. [Right] The function F m,n/f α) ) is potted to g for some m,n,α). whie arge-g expansion is given by Fg) = πg /2 4 g g 2 5 ) 768 g 3 +Og 4 ). 4.4) Notice that the sma-g expansion has a Stokes ine at argg) = 0, π and anti-stokes ines at argg) = ±π/2 as summarized in fig. 9. We can again understand this from the viewpoint of sadde points anaysis. Sadde points of the integration are given by x = 0,±π/2. At the sadde points, the action Sx) = 2g sin2 x takes the vaues Sx = 0) = 0, Sx = ±π/2) = 2g. 4.5) We can pick up a the sadde points through steepest descent except 2 for argg) = 0. We again have reative minus sign in the contributions from the non-trivia sadde points x = ±π/2 because directions of the steepest descent through x = ±π/2 are opposite between the cases for 0 < argg) < π and π < argg) < 0. As in the 0d ϕ 4 theory, beow we consider interpoating functions aong the positive and negative rea axis, and study their anaytic properties as compex functions. 4. Interpoation aong positive rea axis We start with interpoating functions aong the non-negative rea axis of g and then anayticay continue to compex couping. For this case, we have the foowing sma-g and arge-g expansions F Ns) s g) = N s k=0 s k g k, s k = 2π 2k Γ 2 k +/2) Γk +)Γ 2 /2), 2 Note that for argg) = 0 ±, direction of the steepest descent around x = ±π/2 is argx) = π/2 0 ± ) whie the one for x = 0 is argg) = 0 ±. 5

17 Figure : [Left] In the shaded region, the ratio F /2) 8,8 /F is arger than We aso pot zeros +) and poes ) of F /2) 8,8 g). The bue soid ine denotes the branch cuts. [Right] First poes and zeros of F m,n /2) ) 2 from the positive rea axis for various m,n). F N ) g) = g /2 N =0 k g k, k = Γk +/2) π Γk +), 4.6) which are compared with the exact resut in fig. 0 [Left]. In terms of these expansions, we can construct interpoating function F m,ng) α) see app. C.2 for expicit forms). In fig. 0 [Right], we compare some FPRs with the exact resut for argg) = 0. As in the 0d ϕ 4 theory, we find that these interpoating functions we approximate the exact resut for positive rea g. For exampe, the best approximation F /2) 8,8 g) among the FPRs has O0.%) error at worst. Next et us consider compex g in the best FPR F /2) 8,8 g). We summarize the vaidity of the approximation with anaytic structures of F /2) 8,8 g) in fig. [Left]. The shaded region again starts with the first branch cuts seen from the positive rea axis, which are given by the first poes and zeros of the rationa function F /2) 8,8 g)) 2. The first poes are ocated at argg) ±0.9348, which is not near from the anti-stokes ines at argg) = π/2 ±.57. We again expect that the first branch cuts wi vanish as increasing m,n) in F m,n /2) g). Indeed fig. [Right] impies that the first poes wi coapse to the first zeros for arge m,n) as in sec Hence we concude that the branch cuts are the fake singuarities and the FPR F m,n /2) g) with sufficienty arge m, n) woud give good approximation in the right-haf pane of g. In next subsection, we aim to construct interpoating functions approximating the exact resut on the eft-haf pane by considering interpoating probem aong the negative rea axis. 6

18 Figure2: [Left] The function Fg) = ie 2g Fg)back soid), its sma-g expansions FN s) s g) red dotted) and arge-g expansions F N ) g) bue dashed) for argg) = π ǫ. [Right] The reative error F m,n/ F α) ) is potted to g for rea negative g. 4.2 Interpoation aong negative rea axis Let us consider interpoating probem aong the negative rea axis. The function Fg) has the foowing expansions π π F t+iǫ) = t+9t2 +Ot 3 )) i 32 e 2t 8+4t 9t 2 +Ot 3 ))+Oǫ) = it /2 π + 4 πt πt ) 768 πt 3 +Ot 4 ) +Oǫ), π π F t iǫ) = t+9t2 +Ot 3 ))+i 32 e 2t 8+4t 9t 2 +Ot 3 ))+Oǫ), = +it /2 π + 4 πt πt ) 768 πt 3 +Ot 4 ) +Oǫ), 4.7) with t R +. The dominant parts of the expansions change the signs across the negative rea axis as in the ϕ 4 theory since Fg) has the branch cut on the rea negative axis. Hence it is more appropriate to consider the function The function Ft) has the expansions, Ft) = 2π + Ft) = im ie g t 2g Fg +iǫ), t R ) ǫ +0 π 2 t+ 9 4 π 2 t π 8 2 t3 +Ot 4 ) = t /2 π 4 πt πt πt 3 +Ot 4 ) ), 4.9) where we have dropped the exponentiay suppressed correction Oe 2t) in the sma-g expansion see fig. 2 [Left] for comparison of these expansions with the exact resut of Ft)). α) Then we can construct the FPR F m,nt) to interpoate these expansions and approximate the origina function Fg) by F α) L±,m,n g) = ie 2g Fα) m,n g). 4.0) 7

19 Figure 3: [Left] The region where the ratio F /2) L+,9,9 /F is arger than We aso pot zeros +) and poes ) of the rationa function F 0,0) /2) 2. The bue soid ines denote the branch cuts of F /2) L+,9,9. [Right] Simiar pot as the eft for F /2) L,9,9. ThefunctionF α) L±,m,n g)reproduces thesma-g andarge-g expansions offg)forg R ±iǫ up to certain orders. α) In fig. 2 [Right] we check that the interpoating functions F m,ng) we approximate Fg) on the negative rea axis. We again summarize the vaidity of approximation by F /2) L±,9,9 g) and its anaytic property in fig. 3. Starting with the negative rea axis countercockwise cockwise), theapproximation F /2) L+,9,9 g)f /2) L,9,9 g))getsworse acrossthefirst branchcut as in ast subsection, which is ocated around the ine argg) ) and deviates from the anti-stokes ines. However, according to fig. 4 [Left], the first branch cuts seems to shrink as increasing m,n) in F /2) L±m,n g))2. Thus we expect that the branch cut is the fake singuarity and the FPR F /2) L+,m,n g) F /2) L,m,n g)) with sufficienty arge m,n) we approximates Fg) in the eft-top-quarter eft-bottom-quarter) pane. Let us patch the best interpoating functions aong the positive and negative rea axis as in the ϕ 4 theory. In fig, 4 [Right], we draw range of vaidity of approximation by F /2) 8,8 g) for argg) < π/2 F g) = F /2) L+,9,9 g) for π/2 < argg) < π. 4.) F /2) L,9,9 g) for π < argg) < π/2 This indicates that the patching F g) has 5% or better accuracy in the fairy wide region. 5 Concusion and Discussions We have studied anaytic structures of some interpoating functions and discussed their physica impications. We have proposed that the anaytic structures of the interpoating functions 8

20 Figure 4: [Left] First poes and zeros of F /2) L+,m,n )2 from the positive rea axis for various m,n). [Right] The region where the patch F g) of the best FPRs gives more than 5% error. provide information on anaytic property and Stokes phenomena of the physica quantity, which we approximate by the interpoating functions. More concretey, we have mainy considered the roes of the first branch cuts measured from a specific axis where we construct the interpoating functions. If the first branch cuts are not fake singuarities, then we expect that the cuts approximate those of the exact resut or indicate ocations of Stokes or anti-stokes ines. When the cuts are fake singuarities, we shoud consider next cuts in order to get physica impications. We have expicity checked our proposa in the partition functions of the 0d ϕ 4 theory and the Sine-Gordon mode. This seems to hod aso in other exampes such as BPS Wison oop in 4d N = 4 Super Yang-Mis theory, energy spectrum in d anharmonic osciator etc [24]. We expect that our resut is appicabe in more practica probems, where we do not know exact soutions. One possibe appication of our resuts is to use them to find Stokes behavior of the physica quantity by studying anaytic structures of interpoating functions. We have compared the resut of the FPR with the recent resut [20] of the resurgence in the 0d Sine-Gordon mode. We have seen that the finite order approximation of the resurgence give better approximations than the FPR in the region where the FPR gives reativey imprecise approximation whie the FPR is more precise in sufficienty strong couping region. This impies that the FPR and resurgence pay compementary roes. It wi be interesting to compare them in more detai. One of subte points in our approach is concerned with the fake singuarities. It is uncear how we shoud extend the definition of the fake singuarities to more genera probem. We have defined the fake singuarity in our two exampes as the branch cut shrinking for arge m, n) with fixed α. Since the reative errors seem to monotonicay decrease as increasing m,n) in our exampes, it woud be reasonabe to expect that the FPRs with sufficienty arge m, n) have arger range of vaidity in the compex g-pane. However, in genera probem, 9

21 FPRs with arger m, n) do not necessariy give better approximations as seen in [3]. Natura extension of the definition woud be to consider a famiy of FPRs {F m αs) s,n s } satisfying F αs) m s,n s Fg) < F α s+) m s+,n s+ Fg), 5.) and define fake singuarity as a cut, which shrinks as we increase s. It woud be interesting to pursue this direction further. In this paper we have focused on anaytic properties of the best interpoating functions, which provide the best approximation among the given interpoating functions aong a specific axis. However, when we do not know the exact resuts, it is nontrivia which interpoating functions give reativey better approximation as discussed in [3]. Athough the work [3] has proposed a criterion to choose the best interpoating function in terms of two perturbative expansions, we need information on arge order behavior of the expansions to use the criterion. It is nice if we use anaytic properties of the interpoating functions to determine the reativey better interpoating functions without knowing exact resuts. In our exampes, the exact resuts have the square-root type branch cuts. Interestingy the best interpoating functions are at α = ±/2 and hence aso have the square-root type branch cuts. Thus the anaytic properties of the interpoating functions woud be hepfu to improve such criterion. We have seen that each FPR considered here has its own anguar wedge of vaidity. By patching the best FPRs aong the positive and negative rea axis, we have obtained the approximation with arger range of vaidity than each FPR. It woud be interesting if one can construct singe interpoating function, which gives sma-g and arge-g expansions for a argg). One might think that this was conceptuay simiar to finding connection formua between different Stokes domains in exact JWKB method. However there are some important differences. One of such differences is that FPR often sti gives good approximation even across Stokes ine whie approximation by FPR necessariy breaks down across the anti-stokes ine. For exampe, in the 0d ϕ 4 theory, the best FPR aong the positive rea axis gives precise approximation even across the Stokes ine at argg) = π/2 uness we approach to the anti- Stokes ine at argg) = 3π/4. Thus the connection formua in JWKB does not seem to give usefu hints. Let us see one of main difficuties to construct the singe interpoating function vaid for a argg) in the toy exampe: Fg) = F 0 g)+e g F g), with F 0 g) = a 0,k g k, F g) = k=0 a,k g k, k=0 where the summations F 0 g) and F g) are convergent, and the exact resut of Fg) is given by their anaytic continuations. Let us consider FPR to interpoate sma-g and arge-g expansions of Fg). If we construct FPR aong the positive rea axis, then the FPR interpoates F 0 g) and the arge-g expansion of Fg). Some part of the information about the arge-g expansion is encoded in F g), but the FPR does not have this information in sma-g regime. This missing information gives exponentiay suppressed error on the right pane of g and exponentiay growing error on the eft pane for sma g. Simiary if we construct FPR aong the negative rea axis, then that FPR does not have the information that a part of the 20

22 arge-g expansion comes from F 0 g). Of course we can find much better approximation in this exampe by separatey constructing FPRs to interpoate F 0, g) and arge-g expansion of anaytic continuation of F 0, g). However this information is amost equivaent to have the exact resut and there is no motivation to perform this procedure. Acknowedgments We woud ike to thank Ashoke Sen for discussions and usefu suggestions. DPJ woud ike to thank IMSc, Chennai and CHEP, Bengauru for hospitaity during the course of this work. This work was supported in parts by the DAE project 2-R& D-HRI A Comparison with resurgence resut As mentioned in the main text, the authors in [20] have performed resurgence anaysis in the 0d Sine-Gordon mode. It is interesting to compare their resut with our interpoating functions. Their resut is where Fg) = {S argg) Φ 0 g) ie 2g Sargg) Φ g) for argg) 0,π) S argg) Φ 0 g)+ie 2g Sargg) Φ g) for argg) π,0), S θ Φ 0, g) = g + e iθ 0 dt e t g BΦ0, t). A.) A.2) The function BΦ 0 t) BΦ t)) denotes anaytic continuation of Bore transformation of the sma-g expansion coming from the sadde point x = 0 x = ±π/2), which is expicity given by BΦ 0 t) = 2π 2 F 2, ) 2 ;;2t, BΦ t) = 2π 2 F 2, ) 2 ;; 2t. A.3) Athough we know a the coefficients of the sma-t expansions, et us consider finite order approximation of the resurgence resut as in [20] and compare this with the FPRs. Namey, instead of using BΦ 0, t), we terminate its sma-t expansion up to Ot 2N+ ) and use its one-point Padé approximant 3 : P 2N+) t) = N k=0 c kt k + N k= d A.4) ktk, in the integration A.2), which reproduces the sma-t expansion up to Ot 2N+ ). This procedure is often caed one-point) Bore-Padé approximation. In fig. 5, we compare the resut of the resurgence with the FPR F /2) 8,8 for 4 argg) = π. 00 By Og M ) resurgence, we mean the resut obtained by the repacement BΦ 0, t) P M) t) 3 Stricty speaking, this is so-caed diagona Padé approximant. 4 This choice of argg) is due to a technica probem in the integration A.2). As decreasing argg), it becomes harder to obtain precise vaues of the integration. Here we expect that this resut for argg) = π 00 is amost the same as for argg) 0. 2

23 Figure 5: [Left] F /2) 8,8 /F for argg) = π 00 different scae. [Right] The same pot as the eft in a in A.2) and A.). We first find that the resurgence resut gives very precise approximation in weak couping regime. Furthermore a the resuts of the resurgence give better approximations in the region where the FPR gives reativey imprecise approximation. As we go to the arge- g regime, the approximation by the resurgence becomes monotonicay worse but the Og 5 ) resurgence sti gives reasonabe approximation at g = 2. However, if we further go to very arge- g regime, the approximation wi breakdown 5. For exampe, the Og 5 ) resurgence has about 5% error at g = 00 and 40% error at g = 000. Hence in sufficienty strong couping regime, the FPR aways gives the better approximation than the finite order approximation of the resurgence. This resut impies that the FPR and resurgence pay compementary roes in describing the exact function at east in this exampe. It wi be interesting to compare them in more detai and other exampes. B Comments on interpoation in Bore pane When sma-g arge-g) expansion of the function Fg) is convergent, we expect that Fg) is very precisey approximated by the FPRs F m,n α) with arge mn). How about the case where sma-g expansion is asymptotic but Bore summabe? For this case, Bore transformation of the sma-g expansion is convergent. If we construct FPR-type interpoating function in the Bore pane, then one might expect that the Bore-FPR approximates the exact resut of Fg) very we. However, in this appendix, we discuss that the Bore-FPR gives sighty worse approximation than the usua FPR at east for the partition function of the 0d Sine-Gordon mode. We have not understood any cear reasons behind this observation. 5 Ofcourse, if we incude arbitrariyhigher orderterms, then the resurgencecan provide arbitrariyprecise approximation. This is aso the main difference between resurgence method and our interpoating function approach. 22

24 B. Method Let us construct interpoating function for Fg) in Bore pane. For this purpose, it is convenient to introduce foowing quantities: where F Ns) s g) = g M F Ns) s N s N g) = gã s k g k, FN ) g) = g M F N ) g) = g b k g k, k=0 The parameter M is a rea number satisfying ã = M +a, b = M +b. ã, b Ns +. k=0 B.) B.2) B.3) The Bore transformation of the sma-g expansion is B F Ns) s t) = N s k=0 s k Γã+k) tã+k. We aso define action of B to the arge-g expansion as B F N ) t) = N k=0 k Γ b k) t b k. B.4) B.5) Then we can construct the FPR-type interpoating function BF m,nt) α) in Bore pane, which interpoate B F s Ns) t) and B F N ) t) up to ) and Ot b N Otã+Ns 2 ), respectivey: BF α) m,n Ns) t) B F s t) = ), BF α) Otã+Ns m,n t) B F N ) t) = Ot b N 2 ). B.6) This impies that the origina function Fg) is approximated by F Bα) m,n = g M 0 dt e t g BF α) m,n t), B.7) which we ca Bore FPR. Indeed one can show that the function Fm,n Bα) gives F s Ns) g) in sma-g regime and F N ) g) in arge-g regime up to some orders 6. One of subte points in this approach is that different vaues of M give different Fm,n Bα) even if we consider the same m,n,α). In this paper, we wi not study M-dependence and fix as M = 0. B.2 Comparison with usua interpoating function in 0d Sine-Gordon mode Let us compare the Bore FPR with the usua FPR in the 0d Sine-Gordon mode for argg) = 0. In fig. 6 we pot F 20) s F, F F 20) F, F F /2) m,n F, F F B /2) m,n F F, 6 If M did not satisfy the condition B.3), then we coud not guarantee this property. 23

25 Figure 6: Comparison of the Bore FPR Fm,n Bα),),2,2),3,3),4,4) with α,m) = /2,0). and usua FPR F α) m,n for m,n) = against g for some vaues of m,n). We find that the Bore FPRs give sighty worse approximations than the usua FPRs with the same m,n,α) at east for these four cases. This differs from our naive expectation and we have not found any cear reasons for that. We wi not address this issue further, but it woud be interesting to find some good interpretation or modification of our construction to get better approximation than the usua FPR. C Expicit forms of interpoating functions In this appendix, we write down expicit forms for interpoating functions used in the main text. C. Partition function of 0d ϕ 4 theory C.. Interpoation aong positive rea axis of g F /2) g)= ) /2 8πg 2π 0,0 Γ/4) 2 +, F /2) 8πgΓ/4)+Γ/4) 3 ) /2 +2πΓ /4) g)= 2πΓ/4), 64π 2 g 2 +8πgΓ/4) 2 +Γ/4) 4, +2πΓ /4)Γ/4) F /6) ) /6, g)= g g 2, F /2) 37.97g + 2,2 g)= g g g g+, F /0) 2,2 g)= g g g 3 +30g 2 +) /0, F /2) g g)= g g+ 3, g g g g+, F /6) g)= , g g g g g g g+ )/6, C.) 24

26 F /4) g)= g 3, g g g 4 +42g 2 +) /4, F /2) g)= ,4 F /0) [ 4,4 g)= g g g g g g g g g+, g g g g g g g g g+ ]/0, F /8) g)= g 4, g g g g g 4 +54g 2 +) /8, F /2) g )g g)= g )g g ) 5,5 g g )g g )g g ), F /2) g g)= g )g g )g g ) 6,6 g )g g+0.029)g g )g g ), F /26) g)= , g g g g g g g g g g g g g+) 26. C.2) C..2 Interpoation aong negative rea axis of g F /2) 0,0 t)=2π 8tΓ 4) 3 2, F/2) Γ 5 4) 8tΓ 4) 3 2 ) +π, t)=2π +π Γ 5 4) 64t 2 Γ 4) πtΓ 3 4) 2 ) +π 2 F /2) 2,2 t)= t t t t t , F/2) 3,3 t)=.8 F /2) 4,4 t)=.8 t t t t t t t t t+0.006, F /2) 5,5 t)=.8 F /2) t)=.8 6,6 t+0.30)t 2 0.9t+0.044)t t+0.050) t t+0.063)t t+0.076)t t+0.037), t t+0.037)t t+0.036)t t+0.07) t+0.8)t t+0.050)t t+0.07)t t+0.042), F /2) t+0.3)t t+0.04)t t+0.05) t)=2. 7,7 t 2 0.t+0.06)t 2 +0.t+0.05)t t+0.03), F /2) t)=.8 8,8 F /2) t)=.8 9,9 +2Γ 3 4) 3 t 2 0.3t+0.03)t t+0.03)t t+0.02)t t+0.06) t+0.6)t 2 0.3t+0.04)t 2 0.4t+0.05)t t+0.05)t t+0.02), /2 +2πΓ 3 4) 3 t+0.398)t t ) t t+0.05)t t ), t+0.25)t t+0.026)t 2 0.t+0.028)t t+0.026)t t+0.05) C.3) t t+0.03)t 2 0.6t+0.04)t t+0.05)t t+0.034)t t+0.033), C.2 Partition function of 0d Sine-Gordon mode C.2. Interpoation aong positive rea axis of g F /2), g)= 2π 22+π)g 2 +3πg+π )π 2+π)g+π ) /2, F /2) 2,2 g)= 2π F /2) g g+0.22)g 2 ) / g ) g)= ,3 g )g 2, g ) 328+5π+5π 2 )g 3 +8π9+8π)g 2 +3π37π 28)g+2π0 35π+6π 2 ) 68+5π+5π 2 )g π+32π 2 )g+32π 2 70π+20 F /2) g )g g)= g )g 2 ) / g ) 4,4 g g )g 2, g ) F /2) g g )g g )g 2 ) / g ) g)= ,5 g )g g )g 2, g ) F /2) g )g g)= g )g g )g 2 ) / g ) 6,6 g g )g g )g 2, g ) F /2) g g )g g )g g )g 2 ) / g ) 7,7 g)=3.459 g )g g )g g )g 2, g ) F /2) g )g g)= g )g g )g g )g 2 ) / g ) 8,8 g g )g g )g g )g g ) C.4) C.2.2 Interpoation aong negative rea axis of g F /2), t)= 2π 22+π)t 2 +3πt+π )π 2+π)t+π ) /2, F /2) 2,2 t)= 2π 328+5π+5π 2 )t 3 +8π9+8π)t 2 +3π37π 28)t+2π0 35π+6π 2 ) 68+5π+5π 2 )t π+32π 2 )t+32π 2 70π+20, ) /2, ) /2, F /2) t t+0.22)t 2 ) / t ) 3,3 t)=3.459 t )t 2, t ) F /2) t )t t)= t )t 2 ) / t ) 4,4 t t )t 2, C.5) t ) 25