On the Borel-Padé summation of non-convergent series

Bachelor project Natuur- & Sterrenkunde (15EC)

31-3-2014 — 27-6-2014

FNWI

Institute for Theoretical Physics Amsterdam (ITFA)

On the Borel-Pad´

e summation of

non-convergent series

Author:

Stefan Kooij

Student ID:

10243593

Supervisor:

Dr. M.L. Vonk

Second corrector:

Dr. D.M. Hofman

July 11, 2014

1

Summary

In Borel-Pad´e summation of divergent series an appropriate path of integration must be chosen that avoids the poles of the Pad´e approximant. For the perturbative series of the Painlev´e I equation all poles lie on the positive real axis making it easy to determine the right path of integration. It is possible to derive a differential equation from the Quartic integral that contains not only an 1-instanton correction but all orders in its full trans-series solution. The instanton coefficients can be calculated numerically by applying Borel-Pad´e summation on an asymptotic relation that relates the perturbative coefficients with the instanton coefficients. The poles in this calculation have imaginary parts making the search for the right path of integration less straightforward. A simple path yields correct values for some 2-instanton coefficients and zeroth k-instanton coef-ficients. It would be interesting to know if other paths of integration could work and under which conditions they should be used. Further study on, for example, residues and different paths of integration remains to be done.

2

Populair wetenschappelijke samenvatting

Binnen de Natuur- en Wiskunde kom je geregeld als oplossing van een bepaald prob-leem oneindig lange optelsommen tegen. Van sommige van die optelsommen is het niet duidelijk waaraan het gelijk moet zijn. Neem bijvoorbeeld:

Som1= 1 − 1 + 1 − 1 + 1 − 1 + 1 − ...

Of:

Som2= 1 + 2 + 3 + 4 + 5 + 6 + 7 + ... (1)

Som1 is bij het optellen eerst 1 dan 0 en vervolgens weer 1 enz. Wat geeft dit als je

dit oneindig lang blijft optellen? Som2 wordt steeds groter en groter en uiteindelijk

oneindig groot. Natuurkundigen zeggen in beide gevallen dan dat de som divergeert en vaak betekent dit dat er iets niet klopt met de oplossing van je probleem. Maar er zijn verschillende manieren om toch een eindig en eenduidig antwoord te krijgen. Een hele krachtige manier die een eigen naam heeft gekregen, wordt Borel-sommatie genoemd. Aan Som1 blijk je bijvoorbeeld de waarde 1/2 toe te kunnen kennen!

Dit lijkt dan de oplossing van je probleem. Echter bij sommige problemen weet je niet alle termen. Bij Som2 is het simpel de n-de term is gewoon n. Alleen het kan

ook zijn dat je alleen de n-de term kan bepalen als je de vorige termen weet. Vaak wordt het dan steeds moeilijker om volgende termen uit te rekenen en een formule voor alle termen is dan niet te vinden. Helaas blijkt dat je Borel-sommatie alleen kan gebruiken als je zo’n formule voor alle termen hebt. Nu is er een methode om met een eindig aantal termen toch een benadering te maken van het juiste antwoord. Hiervoor schrijf je kort gezegd wat je wilt optellen door middel van Borel-sommatie in ´

e´en van de tussenstappen eerst bij benadering als een breuk. Zo’n benadering wordt een Pad´e benadering genoemd en de gehele methode wordt dan ook Borel-Pad´e sommatie genoemd. Alleen bij ´e´en van de stappen in deze procedure moet de Pad´e benadering ge¨ıntegreerd worden. Maar omdat deze benadering polen heeft, dat zijn plekken waar de functie oneindige waarden aanneemt, moeten deze bij het integreren op de een of andere manier ontweken worden. Dit kan op verschillende manieren en de vraag is hoe dat moet voor een tweetal voorbeelden.

Contents

1 Summary 1

2 Populair wetenschappelijke samenvatting 1

3 Preface 3

4 Introduction 3

5 Painlev´e I 5

5.1 A power series solution . . . 5

5.2 Growth of the coefficients an . . . 6

5.3 Optimal truncation . . . 7

5.4 Trans-series solution . . . 9

5.5 Borel-Pad´e summation . . . 11

6 The quartic integral Z(λ) 14 6.1 A power series solution . . . 14

6.2 F(λ) . . . 15

6.3 Growth of the coefficients bn of F(λ) . . . 16

6.4 Trans-series solution: F(λ) . . . 16

6.5 Borel-Pad´e summation: F(λ) . . . 17

6.6 Borel-Pad´e summation for an asymptotic relation . . . 17

7 Conclusions 19

3

Preface

This report is the end product of a bachelorproject which took place over a period of almost three months. It is the final course of the Bachelor Natuur- & Sterrenkunde. The project is done under the supervision of Dr. M.L. Vonk at the Institute for Theoretical Physics Amsterdam. Since it is a theoretical project of undergraduate level, no signifi-cant new findings have been done, but still some interesting relations were revealed. The objective was to have a good introduction into the subject; Borel-Pad´e summation. Of course this project is more than only a literature review. Proper research was done by working out two examples of non-convergent series, which involves the development of programming to calculate constants of interest and examine their interesting properties.

4

Introduction

At first, divergent series seem useless or a sign of a flaw in the theory. Nevertheless they occur frequently in perturbation theory or as power series solutions of differential equations [1]. Therefore, to make sense out of a divergent power series one must use some mathematical instrument to obtain a finite answer. There exist many so called summation methods. They are not always applicable and the rules involving their validity are complicated and elaborate. So although it would be interesting to investigate the justification of those methods, in this report they are assumed to be valid. A powerful summation method, which is the subject of this report, is Borel summation.

Let f (x) be such a divergent series with coefficients an that grow as n!:

f (x) =X

n≥0

an

xn . (2)

With Stirling’s approximation it can be seen that n! roughly grows as nn_{. From this it}

is clear that for every value of x the factorial growth of the coefficients will eventually always ’win’ from the exponential growth of x, so that indeed the sum will not converge. The Borel transform, ˆf (z), of f (x) is then given by [2]:

ˆ f (z) =X n≥1 an zn−1 (n − 1)!, (3)

where the variable is changed from x to z for clarity reasons. The Borel sum is then equal to:

S(x) = a0+

Z ∞

0

e−xzf (z)dz .ˆ (4)

The power series expansion of this function is equal to that of f (x) in equation (2). For this to be valid it must be possible to analytically continue ˆf (z) to a neighborhood of the positive real axis in a way that the Laplace transform,

Z ∞

0

e−xzf (z)dz ,ˆ (5)

converges. In that case, f (x) is called Borel summable. In the way Borel summation is introduced here, one may be confused why this should work at all. A more intuitively way to view this, is to look at the following identity [1]:

1 = R∞

0 e −t_tn_dt

Multiplying this equality with a divergent series and reversing the order of summation and integration results in Borel summation. Reversing order of summation and integra-tion is not in general a valid thing to do for divergent series, but frequently it gives the right answer.

Example:

An interesting and illustrative example is the sum

f (x) = ∞ X n=0 (−1)n xn , (7)

which diverges whenever |x| ≤ 1. The Borel transform of this power series is ˆ f (z) = ∞ X n=1 (−1)n z n−1 (n − 1)! = − ∞ X n=0 (−1)n z n (n)! = − ∞ X n=0 (−z)n n! = −e −z_{. (8)}

Because the Borel transform is equal to a known analytic function, f (x) is Borel summable. The Borel sum is then

S(x) = 1− Z ∞ 0 e−xze−zdz = 1− Z ∞ 0 e−z(x+1)dz = 1− −e −z(x+1) (x + 1) ∞ 0 = 1− 1 x + 1. (9) Setting x equal to 1, it is possible to conclude that:

∞

X

n=0

(−1)n Borel= 1

2. (10)

This is an interesting result that shows the power of Borel summation.

Throughout the further text another less known type of series appears, namely an asymptotic series. For a Taylor expansion in x the expression becomes more accu-rate for fixed x if more terms of the sum are included. But for an asymtotic series, the expression becomes more accurate for fixed N if x gets smaller. More precisely, a series

∞

X

n=0

anxn (11)

is asymptotic to the function f (x), in the sense of Poincar´e, if [2]:

lim x→0x −N _{f (x) −} N X n=0 anxn ! = 0 ∀N > 0 . (12)

This report focuses on two examples: the non-convergent power series solution of the Painlev´e I equation and a differential equation derived from the Quartic integral. In both examples it would be very difficult or even impossible to use Borel summation directly. The reason for this is that a closed formula for the coefficients of the power series solution is needed. Using only a finite number of terms of the power series is of no use, since Borel summation will just return the original power series (reversing order of summation and integration when a polynomial is involved, is of course a valid thing to do). A powerful way of avoiding this problem is to approximate the polynomial by a fraction of two other polynomials. This is called the Pad´e approximant when the Maclaurin-expansion of the fraction is, up to the nth power, the same as the nth-degree

polynomial it is approximating [1]. When the numerator has degree N and the denom-inator degree M the Pad´e approximant is denoted by P_MN. Using this approximation instead of just the polynomial, finite and very accurate numerical approximations can be achieved.

Since fractions of two polynomials have poles, one should avoid these poles when ap-plying Borel-Pad´e summation. This can be done in many ways and one of the main objectives is to find out what is the most appropriate path of integration.

5

Painlev´

e I

5.1

A power series solution

The Painlev´e I equation is:

u(x)2−1 6u

00_{(x) = x . (13)}

This differential equation is important in many fields, especially in 2D quantum gravity where the free energy of the theory is described by one of its solutions [3]. Inserting the following ansatz: u(x) = x1/2 N X n=0 anx−5n/2. (14)

One obtains using the Cauchy product [4]:

x N X n=0 anx−5n/2 ! N X m=0 amx−5m/2 ! −1 6 d dx N X n=0 an  1 − 5n 2  x(−5n−1)/2 ! = x = x N X n=0 n X m=0 aman−m ! x−5n/2−1 6 N X n=0 an  1 − 5n 2   −1 − 5n 2  x(−5n−3)/2. (15) For this equality to hold the following relation has to apply:

a2₀= 1 =⇒ a0= ±1 . (16)

All powers of x other than x1 must cancel, from this it follows that:

n X m=0 aman−m= 1 6an−1  1 − 5(n − 1) 2   −1 − 5(n − 1) 2  =⇒ n−1 X m=1 aman−m+ 2a0an= an−1 24 25(n − 1) 2_{− 1}
=⇒ an = an−1 48a0 25(n − 1)2− 1
− 1 2a0 n−1 X m=1 aman−m (n > 1) . (17) This is a recursion formula, only a1 must be calculated separately. This can be done by

setting n equal to 1 in the first part of (17): 2a0a1= a0

−1

4 · 6 =⇒ a1= −1

Using the solution a0 = 1 and a1 = −1/48, all further values of an can be calculated with: an = an−1 48 25(n − 1) 2_{− 1
−}1 2 n−1 X m=1 aman−m (19)

With the program Mathematica the coefficients can be calculated with the use of a sim-ple code (see figure 1). It is also possible to let Mathematica try to find a closed form of this recursion formula. Unfortunately Mathematica cannot find any.

In[1]:= nmax = 10; a@1D = -H1  48L;

For@n = 1, n < nmax, n ++, a@n + 1D = H1  48L * H-1 + 25 * HnL ^ 2L * Ha@nDL - H1  2L * HSum@Ha@kDL * Ha@n -k+ 1DL, 8k, 1, n<DLD

Figure 1: Mathematica code for producing the first 10 coefficients an.

Table 1 shows the first six coefficients. From this it seems impossible to guess a possible candidate for a formula for the coefficients.

a0 a1 a2 a3 a4 a5 1 −1 48 −49 4608 −1225 55296 −4412401 42467328 −73560025 84934656 Table 1: The first few coefficients an of the series.

5.2

Growth of the coefficients a

For the coefficients of the Painlev´e I equation, an ∼ A−2n(2n)! for large n [2]. Using

this relation, A can be calculated numerically by defining a new sequence

cn=  |an| (2n)! −1/(2n) . (20) Then to find A: lim n→∞cn= A . (21)

This new sequence cnconverges slowly and since it requires increasingly more calculation

time to produce new coefficients an, it is better to use Richardson extrapolation instead

[1]. In figure 2 the top graph is the normal sequence cn, where it can be seen that it

is not converging very quickly up to the first 800 coefficients. Now if a Richardson-3 extrapolation is done, the resulting sequence already converges much quicker. One can do a second Richardson-3 extrapolation on the newly obtained sequence, which results in an almost horizontal line. This procedure can be done a third time, but for aesthetic reasons the result is left out of the plot of figure 2. This method yields a numerical value for A:

This strongly suggests that A = 8√3/5. But here it must be emphasized that it has not been proven that Richardson extrapolation is a valid procedure for this particular sequence. With the benefit of hindsight it seems valid, since later on this value will be determined exactly in the 1-instanton correction. Furthermore, the plot of figure 2 leaves little doubt that indeed all sequences converge to the same value.

Out[79]= 0 200 400 600 800x 2.76 2.78 2.80 2.82 2.84 y

Figure 2: The upper (orange) line is the sequence cn, the lines below that are (in order)

the first and second Richardson-3 extrapolations.

5.3

Optimal truncation

In asymptotic series the partial sum will first approach the real value and then diverge again [2]. So the problem at hand is to find the partial sum that is closest to the real value. The method of finding this point from which on the sum should be cut off is called optimal truncation. In the case of the solution of the Painlev´e I equation the coefficients grow as (2n)!/A2n_{. With Stirling’s approximation a minimum can be found}

when x is kept constant:

log     constant ∗(2N )! A2N x −5N/2+1/2     ≈ log     constant A2N     + (2N ) ∗ log |2N | − 2N + log   x −5N/2+1/2 . (23)

Differentiating with respect to N and setting this expression equal to zero yields a value for N :

Nopt.truncation=

|Ax5/4_|

2 . (24)

To make this more graphical one can look at the difference between the partial sum with upper limit (N + 1) and N , which is just the (N + 1)th term in the series. The (N + 1)th term can be seen as a first correction on the value of the partial sum (with upper limit N ). This makes it a good indicator for how precise the answer of the optimal truncation method will be. Since these terms grow very rapidly it is better to take the logarithm of

base 10. Of course any bases will work, but base 10 in particularly is very useful since it is related to the number of decimal digits. For example, if the (N + 1)th term is equal to 10−20 _{the logarithm of base 10 of this number is equal to −20, which is apart from}

the minus sign, about the same as the number of accurate decimal places of the partial sum.

See figure 3 for a plot with a value for x that is equal to 30. It is clear there is a minimum around N = 100 and that the expected precision of the optimal truncation method is around 87 decimal places.

Out[21]=

50 100 150 200 250 300 350N

-50 50

Log10@S_HN+1L -S_HNLD

Figure 3: The logarithm of base 10 of the (N+1)th term in the sum.

Equation (24) gives as optimal truncation point N = (4√3 305/4)/5 ≈ 97, 3. Indeed the minimum value is found at N = 97. Later on it can be seen that the actual value of the partial sum (corresponding to N = 97) agrees with the Borel-Pad´e-approximation to many decimal places.

Another way to visualize how the number of terms that are included in the summation affect the accuracy of the solution, is to fill in the partial sum into the right hand side of equation (13), plot the result for various N and compare it with the line y(x) = x. For the result see figure 4.

0 1 2 3 4x 0 1 2 3 4 HuHxLL^2-H16Lu''HxL

Figure 4: Dashed lines are plots of the left hand side of the Painlev´e I equation. Where for the red line 2, for the orange line 3 and the purple line 7 terms of the series are included. The straight line represents y(x) = x.

From figure 4 one may be tempted to conclude, that the less terms are included the better the solution, but this is wrong. It is true that the solution is better in the region where x is small, but worse for greater x. The lines intersect, something that is invisible in the figure.

5.4

Trans-series solution

The solution of the Painlev´e I equation acquired in the previous part is called a ”pertur-bative series”. There also exist ”non-pertur”pertur-bative” solutions, which carry exponentially suppressed terms. These series are characterized by the instanton action A and are called trans-series [2]. The exponential terms are of the form e−A/x and would be invisible in a Taylor expansion, since all Taylor coefficients are zero. Therefore, the trans-series solution should really be seen as an extension of the normal (perturbative) solution. It is no coincidence that the same letter A appears in the growth of the coefficients and as symbol for the instanton action, they are actually the same. To get a qualitative idea why this should be true, the following example will be discussed.

If the growth of the coefficients is an ∼ n!/An, then again it is possible to find a minimum

using Stirling’s approximation: C ∗ exp  log     xNN ! AN      ≈ C ∗ exp  N  log(N ) − 1 + log     xN AN      . (25)

A minimum can be found at N∗= |A/x|. This is the optimal truncation point and to

get a sense what the possible accuracy might be one can look at the difference between the partial sum with upper limit N∗and the partial sum with upper limit N∗+ 1. This

is just equal to the (N∗+ 1)th term:

C ∗ exp (N∗+ 1) log(N∗+ 1) − 1 + log

      ₁ N∗ N∗+1     !! = C ∗ N∗+ 1 N∗ (N∗+1)

exp (−(N∗+ 1)) ∼ exp (−|A/x|) . (26)

So indeed one would expect corrections of the form e−A/x on top of the perturbative solution.

The following equation can be used to find the trans-series solution of the Painlev´e I equation [5]: u(x) ' x1/2 ∞ X n=0 σ₁ne−nA/x−5/4 x−5nβ/4 ∞ X g=0 u(n)_g x−5g/4. (27)

Here σ1 is a free parameter. It is quite clear there must be such a parameter, but that

it can be inserted in this particular way is an ansatz. As can be seen, the trans-series solution is a sum of power series where each power series is multiplied by a power of an exponential. Only the first term of the sum over n has essentially no exponential factor and this is of course the perturbative solution. All other power series are corrections on top of this perturbative solution.

Inserting equation (27) into the Painlev´e I equation one obtains:

x ∞ X n=0 n X m=0 σn1e−nAx 5/4 x−5nβ/4 ∞ X g=0 g X k=0 u(m)_k u(n−m)_g−k x−5g/4+ 1 96 ∞ X n=0 ∞ X g=0 σ₁nu(n)_g e−nAx5/4x(−6−5g−5nβ)/4[4 − 25g2− 25A2_n2_x5/2_{− 25n}2_β2₋ 50 g n(Ax5/4+ β) + 25Anx5/4(1 − 2nβ)] = x . (28)

Taking only the first term in the summation over n will of course return the perturbative solution. This gives the following relations for the constants defined in equation (28):

u(0)₀ = 1 , (29)

u(0)g =

(

ag/2, if g is even ,

0 , if g is odd (g > 0) . (30)

To find the 1-instanton correction one takes the second term in the summation over n of equation (28) and since the first term was already equal to x, this expression must in fact be equal to zero. So for the 1-instanton correction:

2σ1e−Ax 5/4 x1−5β/4 ∞ X g=0 g X k=0 u(0)_k u(1)_g−kx−5g/4+ 1 96 ∞ X g=0 σ1u(1)g e−Ax 5/4 x(−6−5g−5β)/4[4 − 25g2− 25β2_{− 50gβ − 25A}2_x5/2 +(25A(1 − 2β) − 50Ag)x5/4] = 0 . (31)

The highest power of x is (1 − 5β/4) and for that to be equal to zero it follows that: 2 − 25A

2

96 = 0 =⇒ A = ± 8√3

The positive solution for A is indeed the same as the growth factor found in section 5.2. The negative solution is ignored since exponentially suppressed terms are desired, although it would give a separate solution.

The second highest power of x is ((5β − 1)/4), its coefficient is:

2u(1)₁ −25A 2_u(1) 1 96 + 25A(1 − 2β))u(1)₀ 96 . (33)

Filling in the value for A and setting it equal to zero yields: 2(1 − 2β)u(1)₀ = 0 =⇒ β = 1

2. (34)

Of course another solution is u(1)₀ = 0, but then other coefficients would be zero too, which will give an unsatisfying solution. Now it is possible to determine all coefficients u(1)g of the 1-instanton correction in terms of the first coefficient u

(1)

0 . It is most natural

to choose u(1)₀ = 1. Then the recursion formula

u(1)_g+1 = −1 320√3(1 + g) 9u (1) g + 100gu (1) g + 100g 2_u(1) g − 768 g+2 X k=2 u(1)_2+g−ku(0)_k ! (35)

can be found. The derivation is somewhat tedious but fully analogous to the derivation of (19), so therefore only the end result is given.

5.5

Borel-Pad´

e summation

As mentioned in the introduction, another good method besides optimal truncation is Borel-Pad´e summation. Instead of integrating the Borel transform of the partial sum directly, first a Pad´e approximation is made. For an accurate numerical value the first 60 terms of the power series suffice. With Mathematica one can easily make a Pad´e approximation of a polynomial, therefore also of the partial sum that has to be Borel-Pad´e summed. Good results are acquired when both the denominator and numerator of the Pad´e approximation have the same degree, in this case 30. It could be that better results are obtained if the degree of the numerator and denominator are other than equal. Something that can easily be tested but is beyond the scope of this report. Since the Pad´e approximation for the Painlev´e I equation has poles on the positive real axis the path of integration will lie in the complex plane. A possible path is shown in figure 5.

0.5 1.0 1.5 2.0x

-1.0 -0.5 0.5 1.0äy

Figure 5: The 30 poles of the Pad´e approximation and a possible path in the complex plane.

Because the integral converges it is not necessary for the path to come down again on the real axis. For this particularly path the Borel-Pad´e summation for x = 30 yields:

5.4772024245056650211484374864535022260520613... (36) As discussed before the optimal truncation point for x = 30 lies at N = 97. The partial sum then gives:

301/2

N =97

X

n=0

an 30−5n/2' 5.4772024245056650211484374864535022260520613... (37)

These two numbers are the same up to 87 decimal places (more than are displayed here). Notice that such accuracy of the optimal truncation method was expected (See section 5.3). If you would use more terms of the power series the value of the Borel-Pad´e sum, (36), gets more accurate but will differ from the value of the optimal truncation method, (37). It would be interesting to know if using more terms of the power series will always improve precision. Therefore it is necessary to check if the Pad´e approximation of the Borel-Pad´e summation, Pm

m, converges when m → ∞. The problem of the convergence

of the Pad´e approximations ultimately results in the search for all the poles and zeros [1].

A small investigation on the poles of the Pad´e approximation of the Painlev´e I equation reveals some interesting relationships. First of all, the poles, i.e. the zeros of the denominator of the Pad´e approximation (and not at the same time one of the zeros of the numerator), appear to lie on some curve (see figure 6). The best fit for the n poles of for example P80

80 seems to be of the form:

f (n) = a + b

a » -0.00169231 b »0.009102496 fHnL » a + b I1 - n 81M c c »2.07930907 10 20 30 40 50 60 70 nth pole 0.05 0.10 0.15 0.20 0.25 value pole

Figure 6: The poles of P80

80 and a possible fit.

The zeros of the numerator have the same behavior as the zeros of the denominator. The values of the best fit parameters do not give much hope of finding a general formula for the poles. Finding such a formula for the zeros of the numerator and denominator would be quite valuable, since one could easily produce large order Pad´e approximations and investigate its convergence.

Another curious thing is the sum of the value of the poles for each Pm

m (figure 7). Out[30]= 20 40 60 80 100 120 140 m 100 200 300 400 sum

Figure 7: The sum of the values of the poles for each Pm

m from m = 1 to m = 150.

At least up to P250

quite close to the the instanton action A = 8√3/5. A possible explanation is as follows: if one could perform the exact Borel-transform on the Painlev´e I equation one would obtain a function with a singularity at A = 8√3/5. If this would be the only singularity, it is reasonable to think that the poles of the Pad´e approximation should on average be around that value A. This is precisely what figure 7 demonstrates.

A more thorough investigation on the above structures would be very interesting. Un-fortunately, this is beyond the aim of this report.

6

The quartic integral Z(λ)

6.1

A power series solution

The quartic integral, Z(λ), is a function that can be, as the name suggests, given in the form of an integral. Later on, a differential form for Z(λ) will be used. From this another more interesting differential equation will be derived. The reason for this is quite simple: the trans-series solution of Z(λ) is rather dull. It only has a 1-instanton term, while the solution of the newly derived differential equation, F (λ), has infinite. The quartic integral is given by:

Z(λ) = Z

γ

dze−z2/2+λz4/24. (39)

Where the integral runs over some path γ. From now on the path is just the whole real axis, for this makes calculations easier and anything more complicated is unnecessary. Taking the Taylor expansion for the exponential,

ex= ∞ X n=0 xn n! , (40)

the expression for Z(λ) can be rewritten:

Z(λ) = Z ∞ −∞ dze−z2/2 ∞ X n=0 1 n!  λz4 24 n . (41)

If the order of integration and summation is changed this results in:

∞ X n=0 1 n!  λ 24 nZ ∞ −∞ z4ne−z2/2dz . (42)

Here it must be stressed that this procedure is not in general mathematically sound, but the final expression does not give any indication that it isn’t.

Equation (42) can be simplified by using the following standard integrals: Z ∞ −∞ e−ax2dx =r π a, (43) Z ∞ −∞ x2ne−ax2dx = (2n − 1)!! 2n r π a2n+1 =⇒ Z ∞ −∞ x4ne−ax2dx = (4n − 1)!! 22n r π a4n+1. (44)

From this it follows that: Z(λ) ∼√2π + ∞ X n=1 1 n!  λ 24 n_{(4n − 1)!!} 22n √ π24n+1 _=⇒ Z(λ) ∼√2π +√2π ∞ X n=1 (4n − 1)!! n! 1 24nλ n ₍₄₅₎

Z(λ) is an asymptotic series with coefficients that grow factorial. Therefore Borel sum-mation is needed to evaluate Z(λ).

6.2

F(λ)

As stated before the quartic integral itself is not very interesting, so it is now time to derive F (λ). One can easily check that formula 45 is a solution of the following differential equation:

16λ2Z00(λ) + (32λ − 24)Z0(λ) + 3Z(λ) = 0 . (46) The new differential equation is obtained by setting Z(λ) equal to exp[F (λ)]:

3 + (32λ − 24)F0(λ) + 16λ2 F0(λ)2+ F00(λ)
= 0 . (47) Trying a general power series solution,

F (λ) =

∞

X

n=0

bnλβn+ω, (48)

there are (of course) two possible solutions; one with β = 1 and ω = 1 and another with β = 1 and ω = −1. There are other values for β that work, but essentially the solution will be the same as one of the previous two, since just the right coefficients are zero. Because most calculations are comparable with those of the Painlev´e I equation, only the results are given.

Solution 1: F (λ) = ∞ X n=0 bnλn+1, b0= 1 8, b1= 1 12, bn+1= 2 3(n + 2) n−1 X m=0 (m + 1)(n − m)bmbn−1−m+ (2 − n)(n + 1)bn ! (n ≥ 1) . (49) Solution 2: F (λ) = ∞ X n=0 bnλn−1, b0= −3 2 , b1= −1 8 , bn+1= − 2 3n n X m=1 (m − 1)(n − m)bmbn+1−m+ n(n − 1)bn ! (n ≥ 1) . (50)

The coefficients of solution 2 are the same as that of solution 1 except that all odd powers of solution 2 have a minus sign and the fact that solution 2 has an extra term −3/(2x). Later when viewing the trans-series solution it will become more clear where these two solutions came from.

6.3

Growth of the coefficients b

of F(λ)

Similar to the Painlev´e I coefficients the coefficients of F (λ) grow as n!/An _{for large n.}

With the same token as in 5.2 it is possible to determine A numerically. The numer-ical calculation suggests that A is probably equal to 3/2. The trans-series calculation confirms this value.

6.4

Trans-series solution: F(λ)

Using the differential equation for F (λ) the 1-instanton correction is of the form: e−3/2x

∞

X

n=0

cnxn. (51)

Where the coefficients cn are given by:

c0= 1 , c1= − 1 4, cn+1= − 2 3n 3 n−1 X m=0 bm(1 + m)(cn−m+ cn−1−m(−2 + n − m)) + n(n − 1)cn ! . (52)

and bm are of course the perturbation coefficients.

There is an easier way of finding the perturbation coefficients as well as all instanton coefficients. It is possible to use the solution of Z(λ) (equation (45)) to find the solutions of F (λ). The full trans-series solution of the Quartic integral is a linear combination of Z(λ) and e−3/2x_{Z(−λ). Using the linear combination such that the 2π is divided out}

of Z(λ) the solutions of F (λ) can be obtained as follows:

F (λ) = log " 1 + ∞ X n=1 (4n − 1)!! n!  λ 24 n + e−3/2x 1 + ∞ X n=1 (4n − 1)!! n!  −λ 24 n!# = loghf (λ) + e−3/2xf (−λ)i= log  f (λ)  1 + e−3/2xf (−λ) f (λ)  = log [f (λ)] + log  1 + e−3/2xf (−λ) f (λ)  = ∞ X m=1 (−1)m+1 m ∞ X n=1 (4n − 1)!! n!  λ 24 n!m + ∞ X m=1 e−3m/2x(−1) m+1 m ∞ X n=0 (4n − 1)!! n!  −λ 24 n!m, ∞ X n=0 (4n − 1)!! n!  λ 24 n!m . (53)

Where in the last step the Taylor expansion of log[1 + x] was used. It may be possible to work this out further and acquire a formula for the coefficients, but this will probably involve sums of the partition of integers making it not worth the try. By using only a limited amount of terms of the sum it is always possible to find the first coefficients anyway.

In this derivation one could also extract the f (−λ) term from the logarithm thus ob-taining the second solution of the previous section (and its instanton corrections).

6.5

Borel-Pad´

e summation: F(λ)

Just as for the Painlev´e I equation, it is possible to apply Borel-Pad´e summation on F (λ) for the value x = 30 (see figure 8). Notice that, unlike for the Painlev´e I equation, the poles of this Pad´e approximation have imaginary parts. The Borel-Pad´e sum with the path of integration as shown in the figure yields the following value:

−0.3933473896185100837570732.. + i 0.6353914863261371244296330.. (54) This is not a real number which should not be that surprising. It could very well be that the right path of integration is different from the one that is used here. Also no comparison can be made with the optimal truncation method since the optimal truncation point for x = 30 is much smaller than 1 (see section 5.4). A more thorough and useful investigation on this matter is done in the next section.

-5 5 10 15 20 25x -2 -1 1 2 iy

Figure 8: The poles of the Pad´e approximation for F (λ) and some path of integration.

6.6

Borel-Pad´

e summation for an asymptotic relation

In the previous cases some random value for x was summed by using Borel-Pad´e summa-tion. Therefore, the end result was not very applicable. So now Borel-Pad´e summation will be used in a different, more fruitful context.

The perturbation coefficients are related with all instanton coefficients by an asymptotic series. From now on the gth perturbation coefficient is denoted by B(0)g and the nth

k-instanton coefficient by Bn(k). Then in the case of F (λ) this relationship is [5]:

B_g(0)' ∞ X k=1 Sk 1 2πi ∞ X h=0 Γ(g − h + 1) 3 k 2
g−h+1 B (k) h . (55)

Where S1 is called a Stokes constant. Since the first part of the sum over k is of leading

it in terms of S1one finds that for large g the value of S1is very close to √ 2i. Therefore in further calculations: S1= √ 2i.

This has still nothing to do with Borel-Pad´e summation, but if one would like to use equation (55) to find from the perturbation coefficients the instanton coefficients numer-ically, Borel-Pad´e summation becomes necessary. Dividing both sides of equation (55) by (g − b)!, then after some reshuffling and only keeping the leading order terms the bth d-instanton coefficient can be expressed as:

B_b(d)' 3 d 2 g−b+1_{ 2πi} Sd 1  _B(0) g (g − b)!− 3 d 2
g−b+1 Sd 1 d−1 X k=1 Sk 1 3 k 2
g+1 g X h=0 (g − h)! (g − b)! B_h(k) 3 k 2
−h − b−1 X h=0 B_h(d)(g − h)! (g − b)!  3 d 2 h−b . (56)

Where g is some sufficiently big number. The 1-instanton coefficients can be calculated without any trouble, but with higher orders, summations with argument 3 k₂
h

B_h(k)(g − h)!/(g − b)! appear. These summations do not converge as desired, therefore they should be Borel-Pad´e summed. This can be done by first expanding the arguments in powers of 1/g. Something that is most easily done by using the function Series in Mathematica. For example to calculate the zeroth (i.e. b = 0) 2-instanton coefficients, the 1-instanton coefficients must be Borel-Pad´e summed and the expansion in 1/g will then be as follows:

g X h=0 (g − h)! (g)! B(k)_h 3 k 2
−h = 1 − 3(g − 1)! 8(g)! + 9(g − 2)! 128(g)! − 801(g − 3)! 1024(g)! ... = 1 −3 8 1 g + 9 128 1 g2 − 729 1024 1 g3... (57)

It turns out that the poles of these Pad´e approximations have imaginary parts and as pointed out in the introduction an important problem to address is to find which path of integration is appropriate then. Trying to find correlations between the poles is now even more difficult than in the case of the Painlev´e I equation. It seems that the poles stay relatively close to the real axis but there is no way of telling if this remains true for higher orders. From this one could think that integration as in the case of the Painlev´e I might work.

Applying formula (56) in a Mathematica code the path from the origin to (−100i) and then straight to (−100i + ∞) produces the right answer for:

• The first 5 2-instanton coefficients.

• The zeroth k-instanton coefficient for k ∈ {1, 2, 3, 4, 5, 6}.

These statements might be true for more cases but this has not been checked. In calculating the 2-instanton coefficients the real part gives the right answer but it is ac-companied by a big imaginary number. In the case of the zeroth k-instanton coefficients the correct answer is first approached with increasing g, i.e. with more terms of the sum included, but then diverges again. This point from which on it diverges, decreases for higher orders, making this comparable with an asymptotic relation in k.

At least it is now possible to conclude that Borel-Pad´e summation is useful in this context. Unfortunately, something goes wrong, especially for higher order instanton co-efficients. It could be an error in the Mathematica code, but this is not very likely since it produces the right answer for some cases (which involves all pieces of the code). Then some alternation in the path of integration did not help either. Of course alternating the path in a simply connected domain where the Pad´e approximation is holomorphic

is futile for there integration is independent of path [4]. The Pad´e approximation is holomorphic everywhere except at the poles. Changing the path from (−100i) to (100i) could make a difference (although the value of integration is only altered in the sense that it is complex conjugated), and indeed the code produces the wrong answer where it did not do that before. For example in calculating B₁(3), changing integration from below to above the real axis for the 1- and 2-instanton corrections or any variation on this theme does not produce correct answers. Also taking the average of the integrals with integration paths just above and below the real axis, which is actually equivalent with taking the real part, is of no use.

7

Conclusions

Two divergent power series were the subject of this report, namely the Painlev´e I equa-tion and a power series F (λ) derived from the Quartic integral. The former has coeffi-cients that grow as (2n)!/A2n_{where A = 8}√_{3/5 and the latter that grow as n!/A}n_where

A = 3/2. These instanton actions were found both numerically and exactly. Numeri-cally by defining a new sequence which converges to the value A and exactly by finding the trans-series solution. In the numerical calculations Richardson extrapolation proved to be a very useful tool to speed up convergence, although its use has not been justified. The Borel-Pad´e sum of the Painlev´e I equation for x = 30 with a path of integration from the origin to (−100i) and then straight to (−100i + ∞) yields a numerical answer with many significant figures. The optimal truncation method has a similar accuracy and shows the consensus between the two.

The poles of the Pad´e approximations of the Painlev´e I equation are all real-valued and seem to have interesting relations among them. First of all, there is a function that fits the values of the poles remarkably well. Secondly, the values of the poles seem to be on average around the instanton action A = 8√3/5. Further investigation on these correlations was beyond the scope of this report, but it would make a good follow-up project for there remain several unanswered questions. Questions such as:

• Are there other functions that make a good fit as well?

• Is there a formula for the parameters of the fit: making it possible to predict the values of the poles?

• Is the average of the poles really around the instanton action and can this be proven?

For the power series F (λ) the poles of the Pad´e approximant have imaginary parts mak-ing the correct path of integration less straightforward. Usmak-ing Borel-Pad´e summation to find the sum (just as for the Painlev´e I equation) for x = 30 has as result a complex number. This was not investigated any further since for the power series F (λ) Borel-Pad´e summation was used in a more useful context. It would be meaningful though to look at smaller values for x so that the Borel-Pad´e sum could be compared with the value obtained by the optimal truncation method.

As stated above a different application of Borel-Pad´e summation was used for the power series F (λ). Rewriting an asymptotic relation in terms of the instanton coefficients, the instanton coefficients can in principle be calculated from the perturbative coefficients. For this calculation Borel-Pad´e summation is needed to sum the lower order instanton coefficients. Interesting in this case is that the poles of the Pad´e approximation have imaginary parts, making it questionable that a path of integration as one that was used for the Painlev´e I equation will work. It turns out that for the calculations of some 2-instanton coefficients and some zeroth k-instanton coefficients such a path does work.

But especially for higher orders something goes wrong. It might be that there is a flaw in the calculations, but it could also be that another path or even a combination of paths must be used. Several things were tried without success. Therefore some open question are:

• Is there another path of integration that gives correct results?

• Why do the calculations for the 2-instanton coefficients work out, but have such big imaginary parts?

• Could there be a mistake in the calculations?

Altogether the end result is a bit inconclusive. There remains much to be done and many questions to be answered. But those things require time which is limited in a project like this with a fixed time schedule. Hopefully, in the near future, another effort will be made to shed some more light on this subject.

8

Acknowledgments

The subject of this report is not a part of the normal curriculum even though it seems quite essential, making it useful for my further career. So therefore I want to thank my supervisor Dr. M.L. Vonk for introducing me to this wonderful subject and helping me when calculations were not working out as they supposed to be.

References

[1] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer Science+Business Media, LLC, 1999).

[2] M. Marino, (2012), 1206.6272.

[3] P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, Phys.Rept. 254 (1995) 1, hep-th/9306153.

[4] E. Kreyszig, Advanced Engineering Mathematics, 10 ed. (John Wiley & Sons, inc., 2011).

[5] I. Aniceto, R. Schiappa and M. Vonk, Commun.Num.Theor.Phys. 6 (2012) 339, 1106.5922.