A new method to sum divergent power series: educated match
AA new method to sum divergent power series:educated match
Gabriel ´Alvarez
Departamento de F´ısica Te´orica II, Facultad de Ciencias F´ısicas, UniversidadComplutense, 28040 Madrid, SpainE-mail: [email protected]
Harris J. Silverstone
Department of Chemistry, The Johns Hopkins University, 3400 N. Charles Street,Baltimore, Maryland 21218, USAE-mail: [email protected]
Abstract.
We present a method to sum Borel- and Gevrey-summable asymptoticseries by matching the series to be summed with a linear combination of asymptoticseries of known functions that themselves are scaled versions of a single, appropriate,but otherwise unrestricted, function Φ. Both the scaling and linear coefficients arecalculated from Pad´e approximants of a series transformed from the original series byΦ. We discuss in particular the case that Φ is (essentially) a confluent hypergeometricfunction, which includes as special cases the standard Borel-Pad´e and Borel-Leroy-Pad´e methods. A particular advantage is the mechanism to build knowledge aboutthe summed function into the approximants, extending their accuracy and range evenwhen only a few coefficients are available. Several examples from field theory andRayleigh-Schr¨odinger perturbation theory illustrate the method.PACS numbers: 02.30.Lt; 31.15.xp a r X i v : . [ m a t h - ph ] O c t new method to sum divergent power series: educated match
1. Introduction
Summation of divergent asymptotic expansions has led to a vast literature from bothmathematical and physical points of view. The mathematical goal is often to assign astandard sum to a series whose coefficients satisfy certain growth conditions and whosesum satisfies certain conditions at infinity [1, 2]. The physical literature focuses on awide range of specialized, computational methods. Especially since the work carried outin the 1970’s on the coupling constant analyticity of anharmonic oscillators [3, 4, 5], twosummation methods have become dominant: Pad´e approximants and Borel summation.Both have been found useful in fields as diverse as quantum mechanics, statisticalmechanics, quantum field theory, and string theory. Pad´e approximants are mostoften directly used empirically (see, for example, the recent study on the existenceof an ultraviolet zero for the six-loop beta function of the λ Φ theory [6]), and attimes with new, alternative transformation procedures [7]. Borel summability has beenrigorously proved in several instances. The analytic continuation implicit in the Borelsummation process poses a practical problem that has been dealt with in essentially twoways: conformal mappings [8, 9, 10], and Borel-Pad´e approximants. In the latter, theanalytic continuation is again performed empirically by Pad´e approximants of the Borel-transformed series [3, 11, 12, 13]. Most recently, Mera, Pedersen, and Nikoli´c [14, 15]and Pedersen, Mera, and Nikoli´c [16] have developed a method that uses hypergeometricfunctions to sum perturbation theory series using only a few terms.Initially motivated in part by the papers of Mera, Pedersen, and Nikoli´c, we presenthere a new method to build concise, explicit, analytic approximants to the Borel orGevrey sum of an asymptotic power series. These approximants match the series tobe summed with a linear combination of asymptotic series of known functions. Theknown functions are scaled versions of a single function Φ, and both the scaling andlinear coefficients are readily calculated from Pad´e approximants of a transformed seriesdetermined by the original series and by Φ. If Φ is taken to be (essentially) a confluenthypergeometric function, the new method includes as special cases the standard Borel-Pad´e and Borel-Leroy-Pad´e summation methods. Even more important, prior additional(i.e., educated ) knowledge about the summed function can be built into the approximantsvia the function Φ, sometimes dramatically extending the accuracy and range of theapproximants. The “linear combination” here is similar to the linear combination ofthe Janke-Kleinert resummation algorithm, which is described as “re-expanding theasymptotic expansion in a complete set of basis functions”, and which is mathematicallyequivalent to conformal mapping techniques [17]. Our method, in contrast, is essentiallylinked to the theory of Pad´e approximants. new method to sum divergent power series: educated match Φ -Pad´e approximants Our goal is to approximate the Borel sum ψ ( z ) of a divergent power series, ψ ( z ) ∼ ∞ (cid:88) k =0 d k z k , (1)using any appropriate known function Φ( z ) with its own Borel-summable series,Φ( z ) ∼ ∞ (cid:88) k =0 f k ( − z ) k . (2)The method is at the same time hidden in, and a generalization of, the Borel-Pad´esummation method [13], which we briefly review.Let us denote by P n − ( z ) /Q n ( z ) the [ n − , n ] Pad´e approximant of the Boreltransform of the series (1),ˆ ψ B ( z ) = ∞ (cid:88) k =0 d k k ! z k , (3)and let us assume that Q n ( z ) has only simple zeros. Partial fraction expansion, P n − ( z ) Q n ( z ) = n (cid:88) j =1 r j z − z j , (4)and term-by-term integration lead to the standard Borel-Pad´e approximant ψ B , [ n − ,n ] ( z )to ψ ( z ), ψ B , [ n − ,n ] ( z ) = (cid:90) ∞ e − t n (cid:88) j =1 r j zt − z j d t (5)= n (cid:88) j =1 r j − z j E Euler ( − z/z j ) , (6)where we define E Euler ( z ) by E Euler ( z ) = (cid:90) ∞ e − t zt d t = z − e /z E (1 /z ) , (7)and where E (1 /z ) is a standard version of the exponential integral (see chapter 5 ofreference [18]).The two points to note in this derivation are (i) that the E Euler ( z ) in equation (7) isprecisely the Borel sum of the factorially divergent Euler series [19] obtained by setting f k = k ! in equation (2), and (ii) that the asymptotic expansion of ψ B , [ n − ,n ] ( z ) is identicalto that of ψ ( z ) through order z n − , i.e., that n (cid:88) j =1 r j − z j E Euler ( − z/z j ) = n − (cid:88) k =0 d k z k + O ( z n ) . (8)In principle, the 2 n parameters z j and r j could have been determined de nouveau fromequation (8) by substituting in it the Euler series and equating coefficients. new method to sum divergent power series: educated match ψ Φ ( z ) = ∞ (cid:88) k =0 d k f k z k , (9)and the associated new approximants ψ Φ , [ n − ,n ] ( z ) to ψ ( z ) by ψ Φ , [ n − ,n ] ( z ) = n (cid:88) j =1 r j − z j Φ( − z/z j ) . (10)As a generalization of equation (8), the approximants ψ Φ , [ n − ,n ] ( z ), which depend on 2 n parameters r j , z j , ( j = 1 , . . . , n ), satisfy n (cid:88) j =1 r j − z j Φ( − z/z j ) = n − (cid:88) k =0 d k z k + O ( z n ) , (11)and therefore the r j and z j solve the 2 n equations n (cid:88) j =1 r j − z j f k ( z j ) − k = d k , ( k = 0 , , . . . , n − . (12)In practice, these parameters are most easily calculated from the partial fractionexpansion of the [ n − , n ] Pad´e approximant to ˆ ψ Φ ( z ), i.e., P n − ( z ) Q n ( z ) = n − (cid:88) k =0 d k f k z k + O ( z n ) = n (cid:88) j =1 r j z − z j . (13)In other words, the z j are the poles, for simplicity assumed to be simple, and the r j theresidues, of the [ n − , n ] Pad´e approximant to ˆ ψ Φ ( z ). Accordingly we call ψ Φ , [ n − ,n ] ( z )the “[ n − , n ] Φ-Pad´e approximant” to ψ ( z ).The Borel-Pad´e approximant uses no information about the sum ψ ( z ) except forBorel summability. Generally these approximations will not be accurate over thefull range of the variable z . By an “educated” choice of Φ( z ), we mean buildingadditional knowledge about the nature of ψ ( z ) into Φ( z ), which may lead to veryaccurate approximations over the full range of the variable z even when only a verylimited number of coefficients d k of the original asymptotic series are available. Typicalexamples of prior knowledge that can be built into the Φ-Pad´e approximations are thelarge z behavior of ψ ( z ) or perhaps the large k behavior of the coefficients d k . ΦA prime candidate for Φ is the confluent hypergeometric function U (see chapter 13 ofreference [18]) or, more precisely, the functionΦ( z ) = z − a U ( a, a − b, /z ) , (14)for which the coefficients f k in equation (2) are f k = ( a ) k ( b ) k k ! , (15) new method to sum divergent power series: educated match c ) k is defined by ( c ) k = Γ( c + k ) / Γ( c ). Note that thisΦ( z ) is symmetric in a and b , which is more obvious from equation (15) than fromequation (14). From a theoretical point of view the confluent hypergeometric U is anatural choice for at least two reasons. (i) the Borel-Pad´e method is the special case a = b = 1, since z − U (1 , , /z ) = z − e /z E (1 /z ) , (16)which is the E Euler ( z ) of equation (7). (ii) Just as the Borel transform is inverted by theLaplace transform, there is a generalization (which we state without proof) that invertsthe “confluent hypergeometric transform” [see equations (9) and (15)]: ifˆ ψ Φ ( z ) = ∞ (cid:88) k =0 d k k !( a ) k ( b ) k z k , (17)then ψ ( z ) = 1Γ( a )Γ( b ) (cid:90) ∞ ˆ ψ Φ ( zs )e − s s a − U (1 − b, a − b + 1 , s )d s. (18)(When b = 1, U (0 , a, t ) = 1, and the result is the Borel-Leroy transformation [10].)From a practical point of view, the confluent hypergeometric function (14) is also a veryconvenient choice, because as z → ∞ ,Φ( z ) ∼ z − b Γ( a − b )Γ( a ) + z − a Γ( b − a )Γ( b ) , ( a − b (cid:54) = integer) , (19) ∼ z − a log( z ) − γ − ψ (0) ( a )Γ( a ) , ( a = b ) , (20)where γ is Euler’s constant and ψ (0) ( a ) is the polygamma function. Since theapproximant ψ Φ , [ n − ,n ] ( z ) depends linearly on Φ [see equation (10)], an appropriatechoice of a and b permits the large z behavior (if known) of ψ ( z ) to be built intothe Φ-Pad´e approximants. We illustrate these general ideas with several examples andgeneralizations of the method.
3. Examples φ field theroy As the simplest example, the confluent hypergeometric Φ = ( g ) / U (cid:16) , , g (cid:17) triviallysums the perturbative series for the partition function Z ( g ) of zero-dimensional φ theory [9, 10], because Z ( g ) = 1 √ π (cid:90) ∞−∞ e − x / − gx / d x (21)= (3 / (2 g )) / U (3 / , / , / (2 g )) (22)is equal to the Φ of equation (14) with a = 3 / b = 1 / z = 2 g/
3. In fact, the [0 , Z ( g ), Z ( g ) ∼ ∞ (cid:88) k =0 Γ (cid:0) k + (cid:1) Γ (cid:0) k + (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) k ! (cid:18) − g (cid:19) k (23) new method to sum divergent power series: educated match z = − / r = 3 /
2, and is exactly Z ( g ). A second physically relevant example is the Euler-Heisenberg effective Lagrangian [20,21]. For the spinor case in a purely magnetic background, L ( g ) = (cid:90) ∞ e − s/g (cid:18) coth s − s − s (cid:19) d ss , (24)(cf. equations (1.18) and (1.19) in reference [21]), and has the asymptotic expansion, L ( g ) ∼ ∞ (cid:88) k =0 B k +4 (2 k + 4)(2 k + 3)(2 k + 2) (2 g ) k +2 (25) ∼ − g + 4315 g − g + · · · , (26)where B k +4 denote Bernouilli numbers. Standard Borel-Pad´e summation ofequation (25) would involve Pad´e approximants in g that lead to rational functions of s ,i.e., even functions of s , that have to approximate the Borel transform, which is an odd function of s (essentially the non-exponential factor in the integrand of equation (24)).This parity clash can be avoided by takingΦ( z ) = z − U (2 , , /z ) , (27)i.e., a = 2, b = 1, and f k = ( k + 1)! rather than k !. The inverse confluent hypergeometrictransform equation (18) contains the explicit factor s , so that the Φ-transform with a = 2and b = 1 is in fact an even function of s :ˆ L Φ ,a =2 ,b =1 ( s ) = (cid:18) coth s − s − s (cid:19) s . (28)For every n ≥
1, all the poles z j , ( j = 1 , , . . . , n ) , of the [ n − , n ] Pad´e approximantsin s to ˆ L Φ ,a =2 ,b =1 ( s ) are negative and simple, meaning that the poles in s are paired onthe imaginary axis. The resulting approximants have the form, L Φ ,a =2 ,b =1;[ n − ,n ] ( g ) = n (cid:88) j =1 r j − z j (cid:0) Φ(i g/ (cid:112) − z j ) + Φ( − i g/ (cid:112) − z j ) (cid:1) , (29)with the Φ( z ) given by equation (27). For example, the first Pad´e approximant to theΦ-transformed series is − g
45 212 1 g + ∼ − g (cid:18) −
45 4315 13! g + · · · (cid:19) , (30)with z = − , r = − g
45 212 , and the corresponding Φ-Pad´e approximant is L Φ ,a =2 ,b =1;[0 , ( g ) = − g
45 12 (cid:16)
Φ(i g/ (cid:112) /
2) + Φ( − i g/ (cid:112) / (cid:17) . (31)If expanded as a power series in g , this simple approximation reproduces the first twononvanishing terms of equation (26), but at the same time it also captures the functional new method to sum divergent power series: educated match g expansion: in fact L Φ ,a =2 ,b =1;[0 , ( g ) ∼ − (7 /
30) log( g ), while the exactresult is L ( g ) ∼ − (1 /
3) log( g ) [21]. Note that the exact expansion, (cid:18) coth s − s − s (cid:19) s = ∞ (cid:88) j =1 − j π ( j π + s ) , (32)can be viewed as the “[ ∞ − , ∞ ]” Pad´e approximant in s for the Φ-transform, fromwhich the exact poles and residues can be read off: z j = − j π , r j = − j π . (33)With Φ given by equation (27), the resulting Φ-Pad´e infinite sum reproduces L ( g ): L Φ ,a =2 ,b =1;[ ∞− , ∞ ] ( g ) = ∞ (cid:88) j =1 − j π Φ(i g/ ( jπ )) + Φ( − i g/ ( jπ ))2 . (34)We remark in passing that the coefficients − / ( j π ) give the rate of convergence of theapproximants. φ field theory: the quartic anharmonic oscillator Third, we consider one-dimensional φ theory, i.e., the familiar x -perturbed anharmonicoscillator, whose Schr¨odinger equation is (cid:18) − d dx + 12 x + gx (cid:19) Ψ( x ) = E ( g )Ψ( x ) . (35)The first three coefficients of the ground state Rayleigh-Schr¨odinger perturbation seriesare E ( g ) = 12 + 34 g − g + · · · . (36)The coefficients E ( k ) of this Borel-summable [3] series behave like E ( k ) ∼ ( − k +1 / k + π / Γ (cid:18) k + 12 (cid:19) , k → ∞ . (37)More important is the large- g behavior of E ( g ), which follows from a simple scalingargument, E ( g ) ∼ g / ε, as g → ∞ , (38)where ε = 0 . . . . is the ground state energy of the purely quartic oscillator. Ifthe g / behavior is built into Φ, then even a two-parameter [0,1] approximant gives anexcellent fit to E ( g ) all the way from 0 to ∞ . The details are elementary enough toexecute by hand. Because of the sign pattern, we sum the once-subtracted series, ψ ( g ) = E ( g ) − / g , (39)whose large- g behavior is g − / (then multiply by g and add 1/2 to report the results).Equation (19) shows that a suitable Φ with this behavior can be obtained by taking a = 2 / b > a in equation (14). If b were then chosen to fit the exact quartic new method to sum divergent power series: educated match ε , its value would be 0 . . . . . We take b = 1 (Borel-Leroy-Pad´e, but note that a = 2 / ,
1] Pad´e approximantto the transformed series, which needs only the two coefficients 3 / − / E ( g )-series and f = 2 / z = − /
21 and r = 1 /
7. The [0,1]Φ-Pad´e approximant is E Φ , [0 , ( g ) = 12 + 34 (cid:18) (cid:19) / g / U (cid:18) , , g (cid:19) , (40)which, despite its simple origins, turns out to give remarkable agreement with E ( g ) forall g >
0, as seen in Fig. 1. At ∞ , E Φ , [0 , ( g ) ∼ (cid:18) (cid:19) / Γ (cid:18) (cid:19) g / (41)= 0 . . . . g / , as g → ∞ ; (42)the constant 0 . . . . is within 0.4% of the exact quartic ε . Higher-order [ n − , n ]approximants generally agree progressively better. It is clear from Fig. 1 in which the[0,1] Borel-Pad´e approximant is also plotted, how relatively simple information used tochoose the function generating the match can dramatically affect the quality and rangeof the approximant. ‡
20 40 60 80 100 g E ( g ) — E ( g ) ( exact ) — E a = / b = — E Borel - Padé
Figure 1.
Exact E ( g ) (black) and [0,1] approximants for Borel-Pad´e (red) and( a = 2 / , b = 1) confluent hypergeometric Φ (blue). The confluent hypergeometricΦ approximant agrees well with the exact E ( g ), because the g / large- g behavior iscarried by the Φ( g ). As an example of the versatility of the method we show how to incorporate in a simpleway the asymptotic behavior of the coefficients d k into the function Φ. We consider the ‡ All numerical calculations have been done in extended precision using
Mathematica , version 11.1; thecommands, PadeApproximant and HypergeometricU, were particularly relevant. new method to sum divergent power series: educated match β -function for the φ theory in d = 3 dimensions [10], with coefficients˜ β (˜ g ) = 0 − ˜ g + ˜ g − g + 0 . g − . g + 0 . g − . g + O (˜ g ) (43)and growth ˜ β k ∼ ( − . . . . ) k k / k ! , k → ∞ . (44)The [3 ,
4] Pad´e approximant for the Borel transform of ˜ β (˜ g ) has a pole on the positiveaxis at ˜ g = 17 . ,
4] Borel-Pad´e approximant. Stirling’sformula shows that asymptotically the f k in equation (15) go like f k ∼ k ! k a + b − Γ( a )Γ( b ) (cid:18) a − a + b − b + 1 / k (cid:19) , as k → ∞ , (45)so that the growth of the coefficients ˜ β k in equation (44) is matched when a + b = 11 / /k -term is then minimum when a = b = 11 /
4. With this straightforward choice of a and b , and with the corresponding [3 ,
4] approximant to ˜ β (˜ g ), we obtain a value forthe nontrivial root of the β -function of ˜ g ∗ = 1 . Φ -Pad´e approximants for Gevrey-summable series Next we adapt the new Φ-Pad´e approximant method to the cases of summable serieswhose coefficients d k grow like ( mk )!, where m = 2 , , . . . , and which are variously knownas generalized Borel summable [3], m -summable or Gevrey-1 /m summable [2]. The m = 2 case is useful for summing the x -perturbed oscillator and the Euler-Heisenbergseries (25), and m = 3 is useful for the x -perturbed oscillator, etc. We regard theseseries in z with ( mk )! growth to be series in z /m with k ! growth, but in which thecoefficients of all the fractional powers are 0. By averaging over the m -th roots of unity,from a given ( k !)-Φ( z ) [equation (2)] we can construct m appropriate “Gevrey-1 /m ”summed series Φ (1 /m ) µ ( z ), µ = 0 , , . . . , m −
1. Φ (1 /m ) µ ( z ) has the asymptotic series,Φ (1 /m ) µ ( z ) ∼ ∞ (cid:88) k =0 f µ + mk ( − z ) k , (46)and the explicit formula,Φ (1 /m ) µ ( z ) = m (cid:80) mj =1 ω − µjm Φ( − ω jm e π i /m z /m )(e π i /m z /m ) µ , (47)where ω m = e π i /m . The practical procedural consequence is that f k gets replaced by f µ + mk in equations (12) and (13). The question, which µ is appropriate, is similar towhich a and b are appropriate, and the answers depend on which properties, e.g., large z , d k for large k , etc., are most appropriate for ψ . Moreover, the same Gevrey-1 /m Φ (1 /m ) µ new method to sum divergent power series: educated match µ ’s, as illustrated in the nextthree equations and following remark: If, for instance,Φ( z ) ∼ ∞ (cid:88) k =0 k !( − z ) k , (48)then Φ (1 / ( z ) ∼ ∞ (cid:88) k =0 (2 k )!( − z ) k , (49)Φ (1 / ( z ) ∼ ∞ (cid:88) k =0 (2 k + 1)!( − z ) k . (50)The Euler-Heisenberg integral discussed above, particularly equation (29), is betterunderstood as a Gevrey-1/2 series summed by the µ = 0 version of the Φ( z ) given byequation (27), which is the same as the µ = 1 version of z − U (1 , , /z ) [equation (48)]given by equation (50). A classic Gevrey-1/2 series is the Rayleigh-Schr¨odinger perturbation series for the x -perturbed anharmonic oscillator (i.e, the Schr¨odinger equation (35) with gx replacedby gx ). The first three coefficients of the ground-state energy series are E ( g ) = 12 + 158 g − g + · · · . (51)For large k , the coefficients E ( k ) behave like E ( k ) ∼ ( − k +1 (cid:18) π (cid:19) k +1 Γ (cid:18) k + 12 (cid:19) , k → ∞ , (52)and for large gE ( g ) ∼ g / ε, (53)where ε here is the ground-state energy of the pure x oscillator. To build the g / behavior into the approximants, we take (for the once-subtracted series) Φ( z ) = z − / U (3 / , , /z ). Although equation (19) seems to imply that the large- z behaviorwould be z − rather than z − / , the z − term is canceled in constructing Φ (1 / . Whenthe approximant for the subtracted series is multiplied by g , the remaining ( g / ) − / term gives g / . The [0 ,
1] Φ-Pad´e approximant, which like the x case can be done byhand, yields E Φ , [0 , ( g ) = 12 + g
158 Φ (1 / (cid:18) g (cid:19) . (54)This simple [0 ,
1] approximation for the sextic oscillator, while superior to the [0 , n increases the accuracy of the[ n − , n ] Φ-Pad´e approximant increases monotonically to the point that in Fig. 2 it is new method to sum divergent power series: educated match
20 40 60 80 100 g E ( g ) — E ( g ) ( exact ) — E a = / b = [ ] — E Borel - Padé [ ] — E a = / b = [ ] — E Borel - Padé [ ] Figure 2.
Exact E ( g ) (black), Borel-Pad´e approximants (red), and ( a = 3 / , b = 1)-Pad´e approximants (blue) for the x -perturbed anharmonic oscillator. The g / large- g behavior is carried by the ( a = 3 / , b = 1)-confluent-hypergeometric-function-basedΦ (1 / ( g ). The [0,1] and [8,9] approximants are shown. The largest relative error forthe ( a = 3 / , b = 1) [8,9] approximant occurs at g = 100 and is less than 0.007, whichis barely distinguishable from the exact E ( g ). difficult to distinguish between the exact and [8,9]-approximant values for 0 ≤ g ≤ g = 100 is less than 0.007.) The error in the Borel-Pad´eapproximants is much larger.
5. Summary
In summary, the conceptualization presented here emphasizes matching the series tobe summed with a linear combination of asymptotic series of known functions, cf.equation (10). The known functions are scaled versions of a single function Φ( z ), and thescaling and linear coefficients are calculated from the [ n − , n ] Pad´e approximants of thetransformed series generated by Φ( z ). The whole idea stems from the realization thatthe Borel-Pad´e approximant has exactly that structure, but where the Φ( z ) is the sumof Euler’s factorially divergent power series, and from the thought that approximantswould be much more accurate if Φ( z ) were more appropriate for the unknown sum ψ ( z ).Building the long-range behavior of ψ into Φ is particularly successful. Acknowledgments
We wish to acknowledge the support of the Spanish Ministerio de Econom´ıa yCompetitividad under Project No. FIS2015-63966-P and of the Department ofChemistry of the Johns Hopkins University.
References [1] Hardy G H 1949
Divergent series (Oxford: Clarendon) new method to sum divergent power series: educated match [2] Ramis J P 1993 S´eries Divergentes et Th´eories Asymptotiques vol 121 (Marseille: Soci´et´eMath´ematique de France)[3] Graffi S, Grecchi V and Simon B 1970
Phys. Lett. B Ann. Phys. Phys. Lett.
Phys. Rev. D Phys. Rev. D Phys. Rev. Lett. Quantum Field Theory and Critical Phenomena (Oxford: Clarendon)[10] Zinn-Justin J 2010
Appl. Num. Math. Phys. Rev. Lett. Phys. Rev. A J. Phys. A: Math. Gen. Phys. Rev. Lett.
Phys. Rev. B Phys. Rev. A Critical properties of φ -theories (Singapore: WorldScientific)[18] Abramowitz M and Stegun I A (eds) 1970 Handbook of Mathematical Functions (New York: Dover)[19] Euler L 1760 (1754-55)