Tables of the Inverse Laplace Transform of the Function [Formula: see text].

The inverse transform, [Formula: see text], 0 < β < 1, is a stable law that arises in a number of different applications in chemical physics, polymer physics, solid-state physics, and applied mathematics. Because of its important applications, a number of investigators have suggested approximations to g(t). However, there have so far been no accurately calculated values available for checking or other purposes. We present here tables, accurate to six figures, of g(t) for a number of values of β between 0.25 and 0.999. In addition, since g(t), regarded as a function of β, is uni-modal with a peak occurring at t = tmax we both tabulate and graph tmax and 1/g(tmax) as a function of β, as well as giving polynomial approximations to 1/g(tmax).

1. Introduction

It has been known for at least 150 years that mechanical relaxation in solids is non-exponential, the decay often being characterized by a fractional power-law or logarithmic function [1,2]. It is also now generally recognized that all glassy materials exhibit non-exponential relaxation behavior both above and below the glass transition temperature, Tg. This is especially clear from measurements obtained from mechanical [3-6], dielectric [7-9], and photon correlation spectroscopy [10,11]. It is also seen in measurements of volumetric [12], and thermal response [13,14].

In recent years theorists have become interested in the possibility that complex disordered systems exhibit universal features in their relaxation and transport properties, possibly arising from self-similar arrangements of obstacles to motion. This has been particularly encouraged by the observation that nearly all glassy relaxation phenomena can be described by the Kohlrausch-Williams-Watts (KWW) function In many physical applications it is convenient to represent ϕ(t; τ) in the form of a Laplace transform, which we write as where Thus, the function h(v) can be found as an inverse Laplace transform of the function ϕ(t;τ). The function h(v) has found application in the context of the theory of trap-controlled hopping in solid state physics [15,16], chromatography [17], and in the study of models for transport in disordered media [18], as well as in the deconvolution of noisy data [19].

A number of approximate algorithms have been proposed in the literature of chemical physics for the numerical evaluation of h(v) [20-25], in addition to a representation of h(v) in terms of a convergent series given by Pollard [26]. Without loss of generality we can set τ = 1 since h(v) can be represented in terms of the inverse transform where Γ is a line to the right of the origin and parallel to the imaginary axis. The convergent series given by Pollard is In an earlier paper we have presented an accurate tabulation of the sine and cosine transforms of the function exp(− t), needed for the analysis of measurements of dielectric properties taken as a function of frequency [27]. In the present paper we tabulate the inverse Laplace transform h(v). These tables may be used directly for the analysis of experimental data, but are also intended for use as a check on more easily programmed approximations, such as those suggested by earlier investigators [28-30].

2. Numerical Analysis

Two techniques were used to generate the tables that follow which provide an internal check on the accuracy of the computation. The first is that of numerical inversion of the Laplace transform, using a method first suggested by Dubner and Abate [31], and later given in an improved version by Crump [32]. The second is that of direct evaluation of the series given in eq (5). The approximate inverse of a Laplace transform ĝ(s) = ℒ{g(t)} can be expressed in the form of a Fourier series: with an error, E(t) = g(t) − g(t), given by The function can be written in terms of the parameters as In eqs (6) and (7) the constants a and T are arbitrary and can be chosen to maximize accuracy in any particular application. In the present instance, in which , the choice of these parameters is quite straightforward as will be shown below.

Equation (6) was used to evaluate the inverse transform of ĝ(s) for values of β in the range 0.20≤β≤0.999 and values of t ranging from 10−8 (for selected values of β) to 5, to an accuracy of at least nine significant digits. In these ranges of β and t the choice of parametric ranges aϵ(2.5,5) and Tϵ(4,8) sufficed to produce the stated accuracy. The accuracy of the numerical inversion can be checked in detail for three cases in which the inverse transforms are known exactly, where Ai(x) is an Airy function and U(x, y, z) is a confluent hypergeometric function [29]. Typical results for the relative error are given in table 1.

Table 1

Relative errors in the numerical inversion of ĝ(s) for β = 1/3, 1/2, and 2/3 for different values of a and T

	β		1/3	1/2	2/3
a	T	t
2.5	4	0.1	2(−11)	3(−11)	1(−6)
2.5	4	1.0	2(−10)	1(−10)	4(−11)
2.5	4	5.0	6(−10)	5(−10)	4(−10)
2.5	8	0.1	2(−20)	2(−20)	8(−16)
5.0	4	0.01	3(−20)	1(−11)
5.0	4	0.1	4(−20)	7(−20)	3(−15)
5.0	4	1.0	3(−19)	2(−19)	9(−20)
5.0	4	5.0	1(−18)	1(−18)	8(−19)

An alternative approach to the evaluation of g(t) is through the direct series shown in eq (5). The form of the series renders it useful for finding g(t) for large t, but the utility of the series form has occasionally been dismissed because of numerical problems associated with convergence at smaller t. We encountered no difficulties in finding g(t) from eq (5), provided that we used a double precision routine for the gamma functions for k≤22 as well as a Padé correction to Stirling’s approximation at larger k [33]. Thus, we write where The F are constants with the values

3. Tables, Graphs, and Numerical Approximations

The inverse transform of the function ĝ(s) is tabulated in table 2 for the following values of β: 0.25(0.01)0.30(0.02)0.98, 0.99, 0.995, 0.997, 0.998, and 0.999. The finer intervals in β at low values of β are required because of the considerable changes in the function in that neighborhood. Spacings in t vary with β and t in such a way that the peaks of g(t) are most densely covered. There is little need to tabulate g(t) for t > 5 because for these values, the sum of no more than 10 terms of the series in eq (5) suffice to produce g(t) to six-digit accuracy for values of β in the interval (0.05,0.999). For example, if β=0.6 the sum of seven terms of the series gives g(10) to six places, and the sum of four terms gives g(100) to the same accuracy. Figures 1a–c contain graphs of g(t) as a function of t over the entire range of tabulated values of β. Note that for β = 1 g(t) = d(t − 1), a Dirac delta function which is represented as a vertical line in figure 1c.

Table 2

Inverse Laplace transform g(t) of

t	β
	0.25	0.26	0.27	0.28	0.29
0.00001	0.370151D−04	0.515902D−06	0.248661D−08
0.00002	0.151806D−02	0.675577D−04	0.144824D−05	0.123255D−07
0.00003	0.853094D−02	0.649138D−03	0.275324D−04	0.560637D−06	0.451901D−08
0.00004	0.244726D−01	0.258459D−02	0.165894D−03	0.571718D−05	0.902078D−07
0.00005	0.507423D−01	0.672926D−02	0.575158D−03	0.284658D−04	0.711259D−06
0.00006	0.873213D−01	0.137312D−01	0.145291D−02	0.940782D−04	0.330246D−05
0.00007	0.133437D+00	0.239884D−01	0.300022D−02	0.239622D−03	0.109589D−04
0.00008	0.187991D+00	0.376838D−01	0.539788D−02	0.510882D−03	0.289237D−04
0.00009	0.249798D+00	0.548385D−01	0.879472D−02	0.958456D−03	0.647631D−04
0.00010	0.317705D+00	0.753611D−01	0.133042D−01	0.163414D−02	0.128251D−03
0.00011	0.390651D+00	0.990866D−01	0.190057D−01	0.258809D−02	0.231044D−03
0.00012	0.467687D+00	0.125806D+00	0.259484D−01	0.386679D−02	0.386250D−03
0.00013	0.547978D+00	0.155287D+00	0.341556D−01	0.551183D−02	0.607956D−03
0.00014	0.630804D+00	0.187288D+00	0.436295D−01	0.755922D−02	0.910770D−03
0.00015	0.715540D+00	0.221568D+00	0.543553D−01	0.100391D−01	0.130942D−02
0.00016	0.801653D+00	0.257891D+00	0.663051D−01	0.129761D−01	0.181839D−02
0.00017	0.888689D+00	0.296036D+00	0.794409D−01	0.163891D−01	0.245168D−02
0.00018	0.976259D+00	0.335790D+00	0.937176D−01	0.202922D−01	0.322255D−02
0.00019	0.106404D+01	0.376957D+00	0.109085D+00	0.246947D−01	0.414334D−02
0.00020	0.115175D+01	0.419354D+00	0.125488D+00	0.296019D−01	0.522547D−02
0.00021	0.123915D+01	0.462814D+00	0.142873D+00	0.350156D−01	0.647923D−02
0.00022	0.132605D+01	0.507183D+00	0.161181D+00	0.409343D−01	0.791385D−02
0.00023	0.141229D+01	0.552318D+00	0.180357D+00	0.473537D−01	0.953745D−02
0.00024	0.149772D+01	0.598093D+00	0.200343D+00	0.542673D−01	0.113571D−01
0.00025	0.158223D+01	0.644389D+00	0.221084D+00	0.616668D−01	0.133787D−01
0.00030	0.198831D+01	0.880304D+00	0.334310D+00	0.105556D+00	0.266587D−01
0.00035	0.236313D+01	0.111698D+01	0.459197D+00	0.159675D+00	0.453716D−01
0.00040	0.270528D+01	0.134838D+01	0.591057D+00	0.222165D+00	0.694073D−01
0.00045	0.301586D+01	0.157109D+01	0.726376D+00	0.291223D+00	0.984011D−01
0.00050	0.329700D+01	0.178326D+01	0.862582D+00	0.365234D+00	0.131854D+00
0.00060	0.378101D+01	0.217324D+01	0.113082D+01	0.522815D+00	0.209913D+00
0.00070	0.417683D+01	0.251762D+01	0.138668D+01	0.686433D+00	0.299254D+00
0.00080	0.450132D+01	0.281968D+01	0.162607D+01	0.850458D+00	0.396103D+00
0.00090	0.476810D+01	0.308392D+01	0.184753D+01	0.101129D+01	0.497430D+00
0.00100	0.498794D+01	0.331492D+01	0.205105D+01	0.116672D+01	0.600902D+00
0.00110	0.516936D+01	0.351690D+01	0.223731D+01	0.131544D+01	0.704769D+00
0.00120	0.531912D+01	0.369363D+01	0.240740D+01	0.145678D+01	0.807740D+00
0.00130	0.544260D+01	0.384838D+01	0.256249D+01	0.159047D+01	0.908884D+00
0.00140	0.554414D+01	0.398397D+01	0.270380D+01	0.171648D+01	0.100755D+01
0.00150	0.562724D+01	0.410281D+01	0.283253D+01	0.183497D+01	0.110327D+01
0.00160	0.569476D+01	0.420700D+01	0.294978D+01	0.194618D+01	0.119577D+01
0.00170	0.574907D+01	0.429831D+01	0.305657D+01	0.205044D+01	0.128485D+01
0.00180	0.579212D+01	0.437828D+01	0.315385D+01	0.214809D+01	0.137044D+01
0.00190	0.582551D+01	0.444824D+01	0.324247D+01	0.223950D+01	0.145249D+01
0.00200	0.585062D+01	0.450934D+01	0.332320D+01	0.232501D+01	0.153104D+01
0.00210	0.586859D+01	0.456258D+01	0.339676D+01	0.240499D+01	0.160613D+01
0.00220	0.588040D+01	0.460881D+01	0.346376D+01	0.247978D+01	0.167784D+01
0.00230	0.588685D+01	0.464881D+01	0.352477D+01	0.254971D+01	0.174627D+01
0.00240	0.588868D+01	0.468323D+01	0.358032D+01	0.261507D+01	0.181152D+01
0.00250	0.588646D+01	0.471266D+01	0.363086D+01	0.267617D+01	0.187372D+01
0.00300	0.583090D+01	0.480102D+01	0.382187D+01	0.292665D+01	0.214289D+01
0.00350	0.572936D+01	0.482136D+01	0.393647D+01	0.310461D+01	0.235284D+01
0.00400	0.560506D+01	0.480024D+01	0.399971D+01	0.322986D+01	0.251601D+01
0.00450	0.547051D+01	0.475348D+01	0.402773D+01	0.331631D+01	0.264234D+01
0.00500	0.533269D+01	0.469083D+01	0.403120D+01	0.337384D+01	0.273959D+01
0.00600	0.506134D+01	0.454054D+01	0.399133D+01	0.342877D+01	0.286945D+01
0.00700	0.480602D+01	0.437755D+01	0.391576D+01	0.343198D+01	0.293927D+01
0.00800	0.457076D+01	0.421433D+01	0.382270D+01	0.340442D+01	0.296975D+01
0.00900	0.435561D+01	0.405650D+01	0.372195D+01	0.335844D+01	0.297401D+01
0.01000	0.415919D+01	0.390643D+01	0.361892D+01	0.330154D+01	0.296059D+01
0.01100	0.397972D+01	0.376497D+01	0.351666D+01	0.323842D+01	0.293520D+01
0.01200	0.381537D+01	0.363219D+01	0.341686D+01	0.317209D+01	0.290172D+01
0.01300	0.366447D+01	0.350777d+01	0.332044D+01	0.310448D+01	0.286285D+01
0.01400	0.352550D+01	0.339122D+01	0.322787D+01	0.303685D+01	0.282046D+01
0.01500	0.339714D+01	0.328200D+01	0.313931D+01	0.297004D+01	0.277590D+01
0.01600	0.327824D+01	0.317955D+01	0.305479D+01	0.290456D+01	0.273013D+01
0.01700	0.316778D+01	0.308333D+01	0.297421D+01	0.284075D+01	0.268383D+01
0.01800	0.306491D+01	0.299284D+01	0.289744D+01	0.277880D+01	0.263752D+01
0.01900	0.296885D+01	0.290761D+01	0.282430D+01	0.271882D+01	0.259154D+01
0.02000	0.287894D+01	0.282722D+01	0.275460D+01	0.266086D+01	0.254614D+01
0.02100	0.279460D+01	0.275127D+01	0.268816D+01	0.260491D+01	0.250152D+01
0.02200	0.271531D+01	0.267942D+01	0.262478D+01	0.255095D+01	0.245779D+01
0.02300	0.264063D+01	0.261135D+01	0.256428D+01	0.249893D+01	0.241504D+01
0.02400	0.257015D+01	0.254676D+01	0.250649D+01	0.244880D+01	0.237332D+01
0.02500	0.250353D+01	0.248540D+01	0.245125D+01	0.240048D+01	0.233267D+01
0.03000	0.221824D+01	0.221946D+01	0.220816D+01	0.218367D+01	0.214538D+01
0.03500	0.199355D+01	0.200649D+01	0.200950D+01	0.200190D+01	0.198306D+01
0.04000	0.181164D+01	0.183191D+01	0.184418D+01	0.184784D+01	0.184226D+01
0.04500	0.166111D+01	0.168604D+01	0.170445D+01	0.171578D+01	0.171946D+01
0.05000	0.153434D+01	0.156223D+01	0.158475D+01	0.160139D+01	0.161165D+01
0.06000	0.133230D+01	0.136312D+01	0.139020D+01	0.141316D+01	0.143158D+01
0.07000	0.117811D+01	0.120973D+01	0.123868D+01	0.126467D+01	0.128738D+01
0.08000	0.105636D+01	0.108774D+01	0.111719D+01	0.114448D+01	0.116936D+01
0.09000	0.957662D+00	0.988292D+00	0.101751D+01	0.104515D+01	0.107100D+01
0.10000	0.875953D+00	0.905594D+00	0.934205D+00	0.961646D+00	0.987760D+00
0.11000	0.807149D+00	0.835699D+00	0.863499D+00	0.890438D+00	0.916393D+00
0.12000	0.748385D+00	0.775818D+00	0.802711D+00	0.828978D+00	0.854519D+00
0.13000	0.697593D+00	0.723924D+00	0.749875D+00	0.775379D+00	0.800357D+00
0.14000	0.653241D+00	0.678505D+00	0.703514D+00	0.728214D+00	0.752545D+00
0.15000	0.614166D+00	0.638412D+00	0.662499D+00	0.686386D+00	0.710024D+00
0.16000	0.579473D+00	0.602753D+00	0.625950D+00	0.649031D+00	0.671960D+00
0.17000	0.548460D+00	0.570828D+00	0.593171D+00	0.615466D+00	0.637686D+00
0.18000	0.520567D+00	0.542074D+00	0.563604D+00	0.585140D+00	0.606661D+00
0.19000	0.495343D+00	0.516040D+00	0.536798D+00	0.557605D+00	0.578444D+00
0.20000	0.472421D+00	0.492356D+00	0.512382D+00	0.532490D+00	0.552669D+00
0.21000	0.451498D+00	0.470716D+00	0.490048D+00	0.509490D+00	0.529034D+00
0.22000	0.432322D+00	0.450866D+00	0.469541D+00	0.488347D+00	0.507281D+00
0.23000	0.414684D+00	0.432591D+00	0.450645D+00	0.468847D+00	0.487195D+00
0.24000	0.398403D+00	0.415711D+00	0.433176D+00	0.450803D+00	0.468592D+00
0.25000	0.383330D+00	0.400072D+00	0.416980D+00	0.434059D+00	0.451313D+00
0.30000	0.322103D+00	0.336439D+00	0.350960D+00	0.365674D+00	0.380590D+00
0.35000	0.277427D+00	0.289905D+00	0.302564D+00	0.315413D+00	0.328462D+00
0.40000	0.243395D+00	0.254402D+00	0.265577D+00	0.276930D+00	0.288471D+00
0.45000	0.216613D+00	0.226431D+00	0.236401D+00	0.246533D+00	0.256837D+00
0.50000	0.194994D+00	0.203832D+00	0.212807D+00	0.221929D+00	0.231205D+00
0.55000	0.177181D+00	0.185199D+00	0.193341D+00	0.201614D+00	0.210025D+00
0.60000	0.162254D+00	0.169579D+00	0.177013D+00	0.184565D+00	0.192240D+00
0.65000	0.149568D+00	0.156298D+00	0.163126D+00	0.170058D+00	0.177100D+00
0.70000	0.138657D+00	0.144872D+00	0.151174D+00	0.157569D+00	0.164062D+00
0.75000	0.129174D+00	0.134940D+00	0.140783D+00	0.146708D+00	0.152721D+00
0.80000	0.120859D+00	0.126229D+00	0.131668D+00	0.137180D+00	0.142769D+00
0.85000	0.113510D+00	0.118529D+00	0.123610D+00	0.128756D+00	0.133970D+00
0.90000	0.106968D+00	0.111675D+00	0.116437D+00	0.121256D+00	0.126136D+00
0.95000	0.101110D+00	0.105537D+00	0.110012D+00	0.114538D+00	0.119118D+00
1.00000	0.958339D−01	0.100008D+00	0.104226D+00	0.108488D+00	0.112798D+00
1.10000	0.867153D−01	0.904539D−01	0.942257D−01	0.980323D−01	0.101875D+00
1.20000	0.791153D−01	0.824909D−01	0.858919D−01	0.893196D−01	0.927749D−01
1.30000	0.726869D−01	0.757563D−01	0.788447D−01	0.819528D−01	0.850815D−01
1.40000	0.671813D−01	0.699893D−01	0.728108D−01	0.756465D−01	0.784968D−01
1.50000	0.624152D−01	0.649976D−01	0.675892D−01	0.701902D−01	0.728008D−01
1.60000	0.582505D−01	0.606366D−01	0.630282D−01	0.654252D−01	0.678277D−01
1.70000	0.545814D−01	0.567953D−01	0.590116D−01	0.612299D−01	0.634503D−01
1.80000	0.513255D−01	0.533874D−01	0.554483D−01	0.575096D−01	0.595694D−01
1.90000	0.484176D−01	0.503443D−01	0.522683D−01	0.541892D−01	0.561066D−01
2.00000	0.458055D−01	0.476113D−01	0.494124D−01	0.512084D−01	0.529989D−01
2.10000	0.434467D−01	0.451439D−01	0.468347D−01	0.485187D−01	0.501953D−01
2.20000	0.413066D−01	0.429058D−01	0.444971D−01	0.460801D−01	0.476542D−01
2.30000	0.393567D−01	0.408669D−01	0.423681D−01	0.438597D−01	0.453410D−01
2.40000	0.375729D−01	0.390022D−01	0.404214D−01	0.418299D−01	0.432270D−01
2.50000	0.359352D−01	0.372906D−01	0.386350D−01	0.399677D−01	0.412880D−01
2.60000	0.344267D−01	0.357144D−01	0.369902D−01	0.382535D−01	0.395036D−01
2.70000	0.330329D−01	0.342582D−01	0.354710D−01	0.366707D−01	0.378563D−01
2.80000	0.317409D−01	0.329092D−01	0.340640D−01	0.352050D−01	0.363314D−01
2.90000	0.305414D−01	0.316562D−01	0.327573D−01	0.338442D−01	0.349159D−01
3.00000	0.294237D−01	0.304893D−01	0.315408D−01	0.325775D−01	0.335988D−01
3.20000	0.274043D−01	0.283816D−01	0.293441D−01	0.302911D−01	0.312219D−01
3.40000	0.256298D−01	0.265302D−01	0.274154D−01	0.282845D−01	0.291370D−01
3.60000	0.240589D−01	0.248919D−01	0.257093D−01	0.265103D−01	0.272943D−01
3.80000	0.226589D−01	0.234324D−01	0.241901D−01	0.249311D−01	0.256550D−01
4.00000	0.214039D−01	0.221246D−01	0.228292D−01	0.235171D−01	0.241877D−01
4.20000	0.202728D−01	0.209463D−01	0.216036D−01	0.222442D−01	0.228674D−01
4.40000	0.192485D−01	0.198796D−01	0.204945D−01	0.210927D−01	0.216735D−01
4.60000	0.183167D−01	0.189096D−01	0.194863D−01	0.200463D−01	0.205890D−01
4.80000	0.174657D−01	0.180240D−01	0.185661D−01	0.190916D−01	0.196000D−01
5.00000	0.166856D−01	0.172124D−01	0.177231D−01	0.182174D−01	0.186946D−01

Figure 1a

Curves of g(t) as a function of t in the neighborhood of the peak values for β=0.25, 0.275, and 0.30.

Figure 1b

Curves of g(t) as a function of t in the neighborhood of the peak values for β=0.35(0.1)0.95.

Figure 1c

Curves of g(t) as a function of fin the neighborhood of the peak values for β=0.99, 0.995, 0.997, 0.998, and 0.999. The delta function at β=1 is indicated by the vertical line.

It is evident, from the curves shown in figure 1, that the g(t) are unimodal. The position of the peak will be denoted by tmax. Table 3 contains some values of tmax and g(tmax) for the values of β for which we performed our tabulations. It is interesting to observe that among the values of g(tmax) there is a minimum value within the interval (0,1). Figure 2a shows graphs of tmax and 1/g(tmax) as functions of β for values of β between 0.15 and 1. The minimum of g(tmax) occurs at tmax=0.252+ and is equal to 0.888+. These values correspond to β=0.567+. Figure 2b contains a plot of 1/g(tmax) as a function of tmax. Finally, we have derived polynomial least-square approximations to 1/g(tmax) as a function of β. The coefficients of the approximating polynomials as well as a graphical indication of the degree of agreement with our more accurately calculated values of this function are shown in figure 3. A good approximation to 1/g(tmax) probably requires fitting some function other than a polynomial.

Table 3

tmax and g(tmax) as a function of β, where g(t) is the inverse Laplace transform of

β	tmax	g(tmax)
0.15	0.00000190503	450.9129
0.16	0.00000630739	209.22449
0.17	0.0000177337	108.862395
0.18	0.0000436172	62.162073
0.19	0.0000960414	38.311543
0.20	0.000192795	25.155645
0.21	0.000357987	17.416529
0.22	0.000622088	12.610041
0.23	0.00102137	9.484013
0.24	0.00159686	7.369062
0.25	0.00239285	5.888686
0.26	0.00345532	4.821523
0.27	0.00483018	4.032270
0.28	0.00656175	3.435337
0.29	0.00869134	2.974907
0.30	0.0112561	2.613581
0.31	0.0142884	2.325679
0.32	0.0178147	2.093179
0.33	0.0218560	1.903181
0.34	0.0264274	1.746284
0.35	0.0315381	1.615524
0.36	0.0371920	1.505665
0.37	0.0433880	1.412716
0.38	0.0501201	1.333598
0.39	0.0573788	1.265907
0.40	0.0651506	1.207748
0.41	0.0734193	1.157611
0.42	0.0821662	1.114283
0.43	0.0913707	1.076782
0.44	0.101011	1.044308
0.45	0.111063	1.016202
0.46	0.121503	0.991922
0.47	0.132308	0.971016
0.48	0.143453	0.953108
0.49	0.154914	0.937885
0.50	0.166667	0.925082
0.51	0.178689	0.914477
0.52	0.190959	0.905885
0.53	0.203454	0.899150
0.54	0.216155	0.894139
0.55	0.229042	0.890746
0.56	0.242097	0.888881
0.57	0.255304	0.888472
0.58	0.268646	0.889461
0.59	0.282109	0.891806
0.60	0.295680	0.895475
0.61	0.309346	0.900451
0.62	0.323098	0.906726
0.63	0.336924	0.914305
0.64	0.350818	0.923202
0.65	0.364772	0.933445
0.66	0.378780	0.945070
0.67	0.392837	0.958129
0.68	0.406941	0.972686
0.69	0.421088	0.988819
0.70	0.435279	1.006623
0.71	0.449512	1.026212
0.72	0.463791	1.047719
0.73	0.478118	1.071303
0.74	0.492496	1.097151
0.75	0.506933	1.125483
0.76	0.521435	1.156560
0.77	0.536011	1.190689
0.78	0.550671	1.228239
0.79	0.565430	1.269648
0.80	0.580300	1.315445
0.81	0.595300	1.366273
0.82	0.610450	1.422918
0.83	0.625773	1.486349
0.84	0.641295	1.557781
0.85	0.657048	1.638750
0.86	0.673068	1.731228
0.87	0.689397	1.837792
0.88	0.706087	1.961869
0.89	0.723197	2.108126
0.90	0.740799	2.283081
0.91	0.758982	2.496129
0.92	0.777856	2.761327
0.93	0.797561	3.100708
0.94	0.818279	3.550899
0.95	0.840258	4.177648
0.96	0.863851	5.112078
0.97	0.889596	6.659241
0.98	0.918411	9.731437
0.990	0.9522311	18.874343
0.995	0.9725934	37.055102
0.997	0.9819976	61.229747
0.998	0.9871752	91.409296
0.999	0.9928860	181.858807
1.000	1.0000000	INFINITY
1/3	0.0233204	1.847590
2/3	0.388146	0.953613
0.56787475	0.25248543	0.888440006391162 =lowest value of g(tmax)

Figure 2a

Curves of tmax and 1/g(tmax) plotted as a function of β.

Figure 2b

Curve of 1/g(tmax) plotted as a function of tmax.

Figure 3

Second through fourth degree polynomial approximations to 1/g(tmax) as a function of β compared to the more accurately calculated value of this quantity.