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Journal of Research of the National Institute of Standards and Technology logoLink to Journal of Research of the National Institute of Standards and Technology
. 1990 Jul-Aug;95(4):433–467. doi: 10.6028/jres.095.036

Tables of the Inverse Laplace Transform of the Function esβ

Menachem Dishon 1, John T Bendler 2, George H Weiss 3
PMCID: PMC4922407  PMID: 28179785

Abstract

The inverse transform, g(t)=L1(esβ), 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).

Keywords: numerical inversion of Laplace transforms, relaxation processes, stable laws, stretched exponentials

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 [36], dielectric [79], 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

ϕ(t;τ)=exp[(tτ)β],0<β<1. (1)

In many physical applications it is convenient to represent ϕ(t; τ) in the form of a Laplace transform, which we write as

ϕ(t;τ)=0ρτ(u)etudu=0evthτ(v)dv (2)

where

hτ(v)=ρτ(1v)v2. (3)

Thus, the function hτ(v) can be found as an inverse Laplace transform of the function ϕ(t;τ). The function hτ(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 hτ(v) [2025], in addition to a representation of hτ(v) in terms of a convergent series given by Pollard [26]. Without loss of generality we can set τ = 1 since hτ(v) can be represented in terms of the inverse transform

hτ(v)=τ2πiΓevτssβds (4)

where Γ is a line to the right of the origin and parallel to the imaginary axis. The convergent series given by Pollard is

h1(v)=1πk=1(1)k+1Γ(βk+1)sin(πβk)k!vβk+1. (5)

In an earlier paper we have presented an accurate tabulation of the sine and cosine transforms of the function exp(− tβ), 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 hτ(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 [2830].

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) = {g(t)} can be expressed in the form of a Fourier series:

ga(t)~eatT(g^(a)2+k=1[Re{g^(a+ikπT)}cos(kπtT)Im{g^(a+ikπT)}sinkπtT]) (6)

with an error, E(t) = ga(t) − g(t), given by

E(t)=n=1e2naTg(2nT+t). (7)

The function g^(a+ikπT) can be written in terms of the parameters

bk=kπ/T,rk=a2+bk2,θk=tan1(bk/a)

as

g^(a+ikπT)=exp[rkβcos(βθk)][cos(rkβsin(βθk))isin(rkβsin(βθk))]. (8)

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 g^(s)=exp(sβ), 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 (2.5,5) and (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,

β=1/3:g(t)=1(3t4)13Ai[1(3t)13]
β=1/2:g(t)=exp(14t)2π1/2t3/2 (9)
β=2/3:g(t)=243exp(427t2)33/2π1/2t7/3U(16,43,427t2)

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

Γ(1+βk)k!tβk+1~egP(βk)P(k) (10)

where

g=k(1β)+(βk12)ln(βk)(k12)ln(k)(βk+1)ln(t)P(k)=i=05Fiki. (11)

The Fi are constants with the values

F0=1,F1=112,F2=1288,F3=13951,840,F4=5712,488,320,andF5=163,879209,018,880.

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 1ac 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 g^(s)=esβ

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
t β
0.30 0.32 0.34 0.36 0.38
0.0001 0.577115D−05 0.121648D−08
0.0002 0.646619D−03 0.239900D−05
0.0003 0.513202D−02 0.651487D−04 0.107344D−06
0.0004 0.173393D−01 0.450932D−03 0.224946D−05
0.0005 0.392873D−01 0.165274D−02 0.171861D−04 0.212894D−07
0.0006 0.711617D−01 0.424768D−02 0.750118D−04 0.210542D−06
0.0007 0.112072D+00 0.875023D−02 0.231320D−03 0.120478D−05
0.0008 0.160657D+00 0.155375D−01 0.565412D−03 0.478961D−05 0.438912D−08
0.0009 0.215447D+00 0.248343D−01 0.117278D−02 0.147371D−04 0.248393D−07
0.0010 0.275045D+00 0.367290D−01 0.215567D−02 0.375741D−04 0.104716D−06
0.0011 0.338203D+00 0.512025D−01 0.361447D−02 0.830781D−04 0.353443D−06
0.0012 0.403853D+00 0.681562D−01 0.564119D−02 0.164407D−03 0.100396D−05
0.0013 0.471095D+00 0.874382D−01 0.831519D−02 0.297882D−03 0.248718D−05
0.0014 0.539192D+00 0.108863D+00 0.117010D−01 0.502496D−03 0.551702D−05
0.0015 0.607539D+00 0.132228D+00 0.158477D−01 0.799254D−03 0.111767D−04
0.0016 0.675651D+00 0.157324D+00 0.207892D−01 0.121045D−02 0.209968D−04
0.0017 0.743142D+00 0.183942D+00 0.265457D−01 0.175892D−02 0.370175D−04
0.0018 0.809706D+00 0.211881D+00 0.331246D−01 0.246740D−02 0.618297D−04
0.0019 0.875106D+00 0.240950D+00 0.405229D−01 0.335793D−02 0.985926D−04
0.0020 0.939159D+00 0.270970D+00 0.487283D−01 0.445136D−02 0.151029D−03
0.0021 0.100173D+01 0.301776D+00 0.577210D−01 0.576702D−02 0.223396D−03
0.0022 0.106272D+01 0.333216D+00 0.674750D−01 0.732241D−02 0.320443D−03
0.0023 0.112206D+01 0.365153D+00 0.779600D−01 0.913304D−02 0.447353D−03
0.0024 0.117970D+01 0.397460D+00 0.891418D−01 0.112123D−01 0.609668D−03
0.0025 0.123562D+01 0.430026D+00 0.100984D+00 0.135716D−01 0.813220D−03
0.0030 0.148946D+01 0.593555D+00 0.168732D+00 0.298190D−01 0.266390D−02
0.0035 0.170271D+01 0.752371D+00 0.247165D+00 0.536557D−01 0.645439D−02
0.0040 0.188023D+01 0.901685D+00 0.331963D+00 0.846472D−01 0.128304D−01
0.0045 0.202727D+01 0.103934D+Ò1 0.419697D+00 0.121852D+00 0.222298D−01
0.0050 0.214868D+01 0.116469D+01 0.507819D+00 0.164122D+00 0.348631D−01
0.0060 0.233062D+01 0.137960D+01 0.678587D+00 0.259259D+00 0.697247D−01
0.0070 0.245192D+01 0.155178D+01 0.835891D+00 0.361944D+00 0.115960D+00
0.0080 0.253036D+01 0.168824D+01 0.976603D+00 0.466212D+00 0.171035D+00
0.0090 0.257814D+01 0.179553D+01 0.110029D+01 0.568107D+00 0.232237D+00
0.0100 0.260376D+01 0.187922D+01 0.120782D+01 0.665204D+00 0.297120D+00
0.0110 0.261321D+01 0.194384D+01 0.130058D+01 0.756146D+00 0.363668D+00
0.0120 0.261078D+01 0.199304D+01 0.138014D+01 0.840293D+00 0.430308D+00
0.0130 0.259958D+01 0.202974D+01 0.144806D+01 0.917463D+00 0.495866D+00
0.0140 0.258189D+01 0.205627D+01 0.150576D+01 0.987763D+00 0.559498D+00
0.0150 0.255944D+01 0.207450D+01 0.155455D+01 0.105147D+01 0.620622D+00
0.0160 0.253349D+01 0.208594D+01 0.159556D+01 0.110897D+01 0.678861D+00
0.0170 0.250502D+01 0.209182D+01 0.162981D+01 0.116067D+01 0.733992D+00
0.0180 0.247477D+01 0.209311D+01 0.165817D+01 0.120701D+01 0.785906D+00
0.0190 0.244331D+01 0.209064D+01 0.168140D+01 0.124843D+01 0.834581D+00
0.0200 0.241107D+01 0.208507D+01 0.170017D+01 0.128534D+01 0.880053D+00
0.0210 0.237838D+01 0.207695D+01 0.171503D+01 0.131813D+01 0.922401D+00
0.0220 0.234552D+01 0.206673D+01 0.172649D+01 0.134716D+01 0.961733D+00
0.0230 0.231268D+01 0.205478D+01 0.173499D+01 0.137279D+01 0.998177D+00
0.0240 0.228001D+01 0.204143D+01 0.174088D+01 0.139531D+01 0.103187D+01
0.0250 0.224765D+01 0.202693D+01 0.174450D+01 0.141501D+01 0.106296D+01
0.0260 0.221567D+01 0.201150D+01 0.174613D+01 0.143214D+01 0.109158D+01
0.0270 0.218416D+01 0.199533D+01 0.174602D+01 0.144695D+01 0.111789D+01
0.0280 0.215315D+01 0.197858D+01 0.174439D+01 0.145965D+01 0.114203D+01
0.0290 0.212269D+01 0.196137D+01 0.174142D+01 0.147043D+01 0.116412D+01
0.0300 0.209281D+01 0.194381D+01 0.173728D+01 0.147947D+01 0.118430D+01
0.0310 0.206352D+01 0.192601D+01 0.173212D+01 0.148692D+01 0.120269D+01
0.0320 0.203482D+01 0.190803D+01 0.172607D+01 0.149294D+01 0.121942D+01
0.0330 0.200674D+01 0.188995D+01 0.171924D+01 0.149765D+01 0.123458D+01
0.0340 0.197927D+01 0.187183D+01 0.171172D+01 0.150117D+01 0.124830D+01
0.0350 0.195239D+01 0.185370D+01 0.170362D+01 0.150361D+01 0.126065D+01
0.0360 0.192612D+01 0.183561D+01 0.169501D+01 0.150508D+01 0.127175D+01
0.0370 0.190044D+01 0.181760D+01 0.168595D+01 0.150565D+01 0.128166D+01
0.0380 0.187534D+01 0.179970D+01 0.167651D+01 0.150541D+01 0.129048D+01
0.0390 0.185081D+01 0.178192D+01 0.166675D+01 0.150444D+01 0.129828D+01
0.0400 0.182684D+01 0.176430D+01 0.165670D+01 0.150279D+01 0.130512D+01
0.0410 0.180341D+01 0.174684D+01 0.164643D+01 0.150054D+01 0.131108D+01
0.0420 0.178052D+01 0.172956D+01 0.163595D+01 0.149774D+01 0.131621D+01
0.0430 0.175815D+01 0.171247D+01 0.162532D+01 0.149444D+01 0.132058D+01
0.0440 0.173628D+01 0.169558D+01 0.161455D+01 0.149068D+01 0.132422D+01
0.0450 0.171491D+01 0.167890D+01 0.160368D+01 0.148651D+01 0.132720D+01
0.0460 0.169402D+01 0.166243D+01 0.159273D+01 0.148197D+01 0.132956D+01
0.0470 0.167359D+01 0.164618D+01 0.158172D+01 0.147710D+01 0.133134D+01
0.0480 0.165362D+01 0.163015D+01 0.157067D+01 0.147191D+01 0.133258D+01
0.0490 0.163410D+01 0.161435D+01 0.155960D+01 0.146646D+01 0.133332D+01
0.0500 0.161500D+01 0.159877D+01 0.154852D+01 0.146075D+01 0.133359D+01
0.0600 0.144504D+01 0.145517D+01 0.143969D+01 0.139471D+01 0.131684D+01
0.0700 0.130645D+01 0.133216D+01 0.133847D+01 0.132177D+01 0.127837D+01
0.0800 0.119155D+01 0.122661D+01 0.124690D+01 0.124924D+01 0.123015D+01
0.0900 0.109484D+01 0.113550D+01 0.116485D+01 0.118018D+01 0.117840D+01
0.1000 0.101237D+01 0.105627D+01 0.109149D+01 0.111576D+01 0.112640D+01
0.1100 0.941223D+00 0.986854D+00 0.102582D+01 0.105624D+01 0.107582D+01
0.1200 0.879225D+00 0.925600D+00 0.966874D+00 0.100149D+01 0.102752D+01
0.1300 0.824722D+00 0.871187D+00 0.913776D+00 0.951209D+00 0.981859D+00
0.1400 0.776435D+00 0.822554D+00 0.865765D+00 0.905016D+00 0.938950D+00
0.1500 0.733358D+00 0.778843D+00 0.822190D+00 0.862536D+00 0.898755D+00
0.1600 0.694693D+00 0.739355D+00 0.782493D+00 0.823407D+00 0.861163D+00
0.1700 0.659795D+00 0.703513D+00 0.746203D+00 0.787296D+00 0.826024D+00
0.1800 0.628140D+00 0.670841D+00 0.712914D+00 0.753901D+00 0.793174D+00
0.1900 0.599296D+00 0.640941D+00 0.682282D+00 0.722953D+00 0.762444D+00
0.2000 0.572906D+00 0.613478D+00 0.654009D+00 0.694212D+00 0.733674D+00
0.2100 0.548670D+00 0.588169D+00 0.627842D+00 0.667464D+00 0.706710D+00
0.2200 0.526336D+00 0.564774D+00 0.603558D+00 0.642522D+00 0.681408D+00
0.2300 0.505687D+00 0.543084D+00 0.580968D+00 0.619217D+00 0.657637D+00
0.2400 0.486542D+00 0.522922D+00 0.559902D+00 0.597401D+00 0.635274D+00
0.2500 0.468742D+00 0.504133D+00 0.540216D+00 0.576943D+00 0.614211D+00
0.3000 0.395716D+00 0.426630D+00 0.458474D+00 0.491299D+00 0.525137D+00
0.3500 0.341722D+00 0.368917D+00 0.397083D+00 0.426309D+00 0.456686D+00
0.4000 0.300210D+00 0.324328D+00 0.349379D+00 0.375462D+00 0.402687D+00
0.4500 0.267323D+00 0.288884D+00 0.311305D+00 0.334687D+00 0.359140D+00
0.5000 0.240646D+00 0.260059D+00 0.280252D+00 0.301319D+00 0.323364D+00
0.5500 0.218584D+00 0.236178D+00 0.254471D+00 0.273549D+00 0.293507D+00
0.6000 0.200046D+00 0.216084D+00 0.232746D+00 0.250107D+00 0.268253D+00
0.6500 0.184259D+00 0.198954D+00 0.214204D+00 0.230075D+00 0.246642D+00
0.7000 0.170658D+00 0.184187D+00 0.198206D+00 0.212776D+00 0.227960D+00
0.7500 0.158825D+00 0.171331D+00 0.184272D+00 0.197699D+00 0.211668D+00
0.8000 0.148440D+00 0.160045D+00 0.172034D+00 0.184451D+00 0.197345D+00
0.8500 0.139256D+00 0.150062D+00 0.161206D+00 0.172727D+00 0.184667D+00
0.9000 0.131079D+00 0.141172D+00 0.151563D+00 0.162285D+00 0.173373D+00
0.9500 0.123754D+00 0.133208D+00 0.142923D+00 0.152929D+00 0.163256D+00
1.0000 0.117157D+00 0.126035D+00 0.135143D+00 0.144504D+00 0.154145D+00
1.1000 0.105757D+00 0.113641D+00 0.121701D+00 0.129953D+00 0.138414D+00
1.2000 0.962591D−01 0.103319D+00 0.110511D+00 0.117844D+00 0.125330D+00
1.3000 0.882314D−01 0.945984D−01 0.101060D+00 0.107624D+00 0.114296D+00
1.4000 0.813620D−01 0.871394D−01 0.929821D−01 0.988936D−01 0.104877D+00
1.5000 0.754212D−01 0.806920D−01 0.860039D−01 0.913578D−01 0.967540D−01
1.6000 0.702356D−01 0.750676D−01 0.799205D−01 0.847932D−01 0.896840D−01
1.7000 0.656724D−01 0.701212D−01 0.745741D−01 0.790284D−01 0.834810D−01
1.8000 0.616280D−01 0.657397D−01 0.698416D−01 0.739298D−01 0.779998D−01
1.9000 0.580202D−01 0.618337D−01 .0.656257D−01 0.693913D−01 0.731251D−01
2.0000 0.547832D−01 0.583314D−01 0.618482D−01 0.653281D−01 0.687650D−01
2.1000 0.518640D−01 0.551748D−01 0.584460D−01 0.616714D−01 0.648445D−01
2.2000 0.492187D−01 0.523163D−01 0.553672D−01 0.583650D−01 0.613027D−01
2.3000 0.468114D−01 0.497165D−01 0.525690D−01 0.553622D−01 0.580890D−01
2.4000 0.446119D−01 0.473426D−01 0.500157D−01 0.526245D−01 0.551615D−01
2.5000 0.425952D−01 0.451673D−01 0.476776D−01 0.501192D−01 0.524848D−01
2.6000 0.407397D−01 0.431671D−01 0.455291D−01 0.478190D−01 0.500291D−01
2.7000 0.390273D−01 0.413222D−01 0.435489D−01 0.457003D−01 0.477692D−01
2.8000 0.374425D−01 0.396158D−01 0.417184D−01 0.437433D−01 0.456833D−01
2.9000 0.359719D−01 0.380332D−01 0.400218D−01 0.419307D−01 0.437529D−01
3.0000 0.346037D−01 0.365617D−01 0.384453D−01 0.402476D−01 0.419617D−01
3.1000 0.333279D−01 0.351904D−01 0.369770D−01 0.386811D−01 0.402959D−01
3.2000 0.321358D−01 0.339096D−01 0.356065D−01 0.372199D−01 0.387431D−01
3.3000 0.310194D−01 0.327109D−01 0.343246D−01 0.358540D−01 0.372927D−01
3.4000 0.299720D−01 0.315869D−01 0.331232D−01 0.345748D−01 0.359353D−01
3.5000 0.289875D−01 0.305309D−01 0.319952D−01 0.333745D−01 0.346625D−01
3.6000 0.280606D−01 0.295372D−01 0.309344D−01 0.322463D−01 0.334670D−01
3.7000 0.271865D−01 0.286006D−01 0.299350D−01 0.311841D−01 0.323421D−01
3.8000 0.263609D−01 0.277164D−01 0.289921D−01 0.301825D−01 0.312821D−01
3.9000 0.255800D−01 0.268804D−01 0.281011D−01 0.292366D−01 0.302817D−01
4.0000 0.248403D−01 0.260890D−01 0.272580D−01 0.283421D−01 0.293363D−01
4.2000 0.234726D−01 0.246266D−01 0.257013D−01 0.266919D−01 0.275936D−01
4.4000 0.222363D−01 0.233060D−01 0.242970D−01 0.252048D−01 0.260250D−01
4.6000 0.211139D−01 0.221080D−01 0.230243D−01 0.238585D−01 0.246066D−01
4.8000 0.200907D−01 0.210169D−01 0.218662D−01 0.226346D−01 0.233185D−01
5.0000 0.191544D−01 0.200192D−01 0.208083D−01 0.215177D−01 0.221443D−01
t β
0.40 0.42 0.44 0.46 0.48
0.002 0.116562D−05
0.003 0.875664D−04 0.642966D−06
0.004 0.917142D−03 0.216038D−04 0.949439D−07
0.005 0.406492D−02 0.197437D−03 0.259248D−05 0.458884D−08
0.006 0.114034D−01 0.908478D−03 0.249859D−04 0.137126D−06
0.007 0.243242D−01 0.277903D−02 0.130277D−03 0.160298D−05 0.250858D−08
0.008 0.434357D−01 0.653168D−02 0.458125D−03 0.103165D−04 0.409163D−07
0.009 0.686419D−01 0.128176D−01 0.123210D−02 0.443480D−04 0.359676D−06
0.010 0.993632D−01 0.221063D−01 0.273747D−02 0.143247D−03 0.204689D−05
0.011 0.134755D+00 0.346471D−01 0.528212D−02 0.375074D−03 0.848024D−05
0.012 0.173873D+00 0.504823D−01 0.915677D−02 0.837893D−03 0.276694D−04
0.013 0.215782D+00 0.694867D−01 0.146046D−01 0.165512D−02 0.750945D−04
0.014 0.259618D+00 0.914136D−01 0.218037D−01 0.296638D−02 0.176300D−03
0.015 0.304618D+00 0.115939D+00 0.308610D−01 0.491645D−02 0.368510D−03
0.016 0.350132D+00 0.142698D+00 0.418152D−01 0.764441D−02 0.700833D−03
0.017 0.395618D+00 0.171312D+00 0.546444D−01 0.112748D−01 0.123301D−02
0.018 0.440637D+00 0.201408D+00 0.692772D−01 0.159112D−01 0.203294D−02
0.019 0.484838D+00 0.232634D+00 0.856035D−01 0.216325D−01 0.317351D−02
0.020 0.527948D+00 0.264664D+00 0.103486D+00 0.284915D−01 0.472905D−02
0.021 0.569763D+00 0.297204D+00 0.122768D+00 0.365153D−01 0.677197D−02
0.022 0.610129D+00 0.329992D+00 0.143285D+00 0.457071D−01 0.936977D−02
0.023 0.648942D+00 0.362802D+00 0.164867D+00 0.560491D−01 0.125826D−01
0.024 0.686132D+00 0.395437D+00 0.187346D+00 0.675050D−01 0.164616D−01
0.025 0.721660D+00 0.427728D+00 0.210559D+00 0.800237D−01 0.210476D−01
0.026 0.755512D+00 0.459537D+00 0.234352D+00 0.935423D−01 0.263709D−01
0.027 0.787691D+00 0.490746D+00 0.258579D+00 0.107989D+00 0.324507D−01
0.028 0.818217D+00 0.521260D+00 0.283106D+00 0.123286D+00 0.392962D−01
0.029 0.847120D+00 0.551004D+00 0.307810D+00 0.139352D+00 0.469066D−01
0.030 0.874438D+00 0.579917D+00 0.332580D+00 0.156103D+00 0.552723D−01
0.031 0.900219D+00 0.607954D+00 0.357315D+00 0.173454D+00 0.643757D−01
0.032 0.924511D+00 0.635082D+00 0.381927D+00 0.191325D+00 0.741927D−01
0.033 0.947368D+00 0.661279D+00 0.406335D+00 0.209633D+00 0.846930D−01
0.034 0.968846D+00 0.686532D+00 0.430472D+00 0.228302D+00 0.958419D−01
0.035 0.989000D+00 0.710833D+00 0.454277D+00 0.247258D+00 0.107601D+00
0.036 0.100789D+01 0.734185D+00 0.477698D+00 0.266430D+00 0.119928D+00
0.037 0.102556D+01 0.756592D+00 0.500691D+00 0.285755D+00 0.132781D+00
0.038 0.104208D+01 0.778065D+00 0.523220D+00 0.305170D+00 0.146115D+00
0.039 0.105750D+01 0.798618D+00 0.545253D+00 0.324619D+00 0.159883D+00
0.040 0.107186D+01 0.818267D+00 0.566765D+00 0.344050D+00 0.174042D+00
0.041 0.108523D+01 0.837032D+00 0.587736D+00 0.363416D+00 0.188546D+00
0.042 0.109765D+01 0.854934D+00 0.608151D+00 0.382673D+00 0.203351D+00
0.043 0.110916D+01 0.871994D+00 0.627997D+00 0.401783D+00 0.218414D+00
0.044 0.111982D+01 0.888237D+00 0.647268D+00 0.420710D+00 0.233694D+00
0.045 0.112967D+01 0.903687D+00 0.665957D+00 0.439423D+00 0.249150D+00
0.046 0.113875D+01 0.918367D+00 0.684064D+00 0.457894D+00 0.264745D+00
0.047 0.114709D+01 0.932303D+00 0.701587D+00 0.476097D+00 0.280442D+00
0.048 0.115474D+01 0.945520D+00 0.718531D+00 0.494012D+00 0.296207D+00
0.049 0.116173D+01 0.958043D+00 0.734898D+00 0.511619D+00 0.312007D+00
0.050 0.116810D+01 0.969895D+00 0.750694D+00 0.528903D+00 0.327812D+00
0.060 0.120401D+01 0.105666D+01 0.879260D+00 0.681859D+00 0.480848D+00
0.070 0.120500D+01 0.109958D+01 0.962365D+00 0.797412D+00 0.614166D+00
0.080 0.118608D+01 0.111388D+01 0.101155D+01 0.879415D+00 0.721629D+00
0.090 0.115610D+01 0.110979D+01 0.103644D+01 0.934203D+00 0.803710D+00
0.100 0.112030D+01 0.109406D+01 0.104425D+01 0.967935D+00 0.863552D+00
0.110 0.108183D+01 0.107110D+01 0.104023D+01 0.985781D+00 0.904981D+00
0.120 0.104258D+01 0.104386D+01 0.102810D+01 0.991828D+00 0.931641D+00
0.130 0.100370D+01 0.101424D+01 0.101054D+01 0.989212D+00 0.946687D+00
0.140 0.965844D+00 0.983542D+00 0.989407D+00 0.980311D+00 0.952723D+00
0.150 0.929393D+00 0.952596D+00 0.966045D+00 0.966908D+00 0.951843D+00
0.160 0.894540D+00 0.921958D+00 0.941400D+00 0.950341D+00 0.945708D+00
0.170 0.861368D+00 0.891987D+00 0.916142D+00 0.931613D+00 0.935623D+00
0.180 0.829885D+00 0.862904D+00 0.890744D+00 0.911470D+00 0.922610D+00
0.190 0.800057D+00 0.834843D+00 0.865538D+00 0.890471D+00 0.907465D+00
0.200 0.771825D+00 0.807874D+00 0.840754D+00 0.869031D+00 0.890808D+00
0.210 0.745114D+00 0.782026D+00 0.816548D+00 0.847460D+00 0.873120D+00
0.220 0.719845D+00 0.757298D+00 0.793022D+00 0.825984D+00 0.854775D+00
0.230 0.695932D+00 0.733672D+00 0.770239D+00 0.804769D+00 0.836061D+00
0.240 0.673294D+00 0.711115D+00 0.748236D+00 0.783936D+00 0.817201D+00
0.250 0.651849D+00 0.689590D+00 0.727027D+00 0.763569D+00 0.798366D+00
0.260 0.631522D+00 0.669052D+00 0.706614D+00 0.743726D+00 0.779685D+00
0.270 0.612240D+00 0.649456D+00 0.686986D+00 0.724445D+00 0.761256D+00
0.280 0.593933D+00 0.630755D+00 0.668127D+00 0.705748D+00 0.743153D+00
0.290 0.576540D+00 0.612903D+00 0.650014D+00 0.687645D+00 0.725428D+00
0.300 0.559999D+00 0.595856D+00 0.632624D+00 0.670140D+00 0.708119D+00
0.310 0.544257D+00 0.579569D+00 0.615929D+00 0.653227D+00 0.691253D+00
0.320 0.529261D+00 0.564002D+00 0.599901D+00 0.636897D+00 0.674844D+00
0.330 0.514965D+00 0.549114D+00 0.584513D+00 0.621139D+00 0.658903D+00
0.340 0.501325D+00 0.534868D+00 0.569735D+00 0.605937D+00 0.643431D+00
0.350 0.488299D+00 0.521229D+00 0.555540D+00 0.591274D+00 0.628428D+00
0.360 0.475851D+00 0.508163D+00 0.541902D+00 0.577132D+00 0.613889D+00
0.370 0.463945D+00 0.495639D+00 0.528793D+00 0.563494D+00 0.599805D+00
0.380 0.452549D+00 0.483627D+00 0.516190D+00 0.550341D+00 0.586169D+00
0.390 0.441633D+00 0.472100D+00 0.504067D+00 0.537653D+00 0.572969D+00
0.400 0.431169D+00 0.461031D+00 0.492402D+00 0.525413D+00 0.560193D+00
0.410 0.421131D+00 0.450396D+00 0.481173D+00 0.513603D+00 0.547829D+00
0.420 0.411496D+00 0.440173D+00 0.470359D+00 0.502205D+00 0.535864D+00
0.430 0.402240D+00 0.430339D+00 0.459941D+00 0.491201D+00 0.524285D+00
0.440 0.393343D+00 0.420875D+00 0.449899D+00 0.480575D+00 0.513078D+00
0.450 0.384786D+00 0.411762D+00 0.440216D+00 0.470312D+00 0.502232D+00
0.460 0.376551D+00 0.402982D+00 0.430874D+00 0.460396D+00 0.491732D+00
0.470 0.368621D+00 0.394518D+00 0.421859D+00 0.450812D+00 0.481566D+00
0.480 0.360979D+00 0.386355D+00 0.413154D+00 0.441546D+00 0.471721D+00
0.490 0.353612D+00 0.378479D+00 0.404747D+00 0.432585D+00 0.462187D+00
0.500 0.346505D+00 0.370874D+00 0.396622D+00 0.423916D+00 0.452950D+00
0.550 0.314454D+00 0.336511D+00 0.359819D+00 0.384538D+00 0.410854D+00
0.600 0.287281D+00 0.307300D+00 0.328438D+00 0.350840D+00 0.374675D+00
0.650 0.263990D+00 0.282217D+00 0.301435D+00 0.321772D+00 0.343378D+00
0.700 0.243835D+00 0.260483D+00 0.278004D+00 0.296508D+00 0.316126D+00
0.750 0.226243D+00 0.241498D+00 0.257517D+00 0.274396D+00 0.292247D+00
0.800 0.210772D+00 0.224794D+00 0.239482D+00 0.254918D+00 0.271198D+00
0.850 0.197073D+00 0.209999D+00 0.223504D+00 0.237658D+00 0.252541D+00
0.900 0.184869D+00 0.196817D+00 0.209268D+00 0.222279D+00 0.235919D+00
0.950 0.173937D+00 0.185010D+00 0.196517D+00 0.208508D+00 0.221036D+00
1.000 0.164093D+00 0.174381D+00 0.185042D+00 0.196118D+00 0.207652D+00
1.100 0.147104D+00 0.156044D+00 0.165256D+00 0.174768D+00 0.184607D+00
1.200 0.132983D+00 0.140815D+00 0.148840D+00 0.157075D+00 0.165535D+00
1.300 0.121084D+00 0.127994D+00 0.135036D+00 0.142215D+00 0.149542D+00
1.400 0.110936D+00 0.117073D+00 0.123290D+00 0.129591D+00 0.135976D+00
1.500 0.102193D+00 0.107673D+00 0.113195D+00 0.118755D+00 0.124352D+00
1.600 0.945906D−01 0.995101D−01 0.104439D+00 0.109371D+00 0.114302D+00
1.700 0.879277D−01 0.923638D−01 0.967832D−01 0.101179D+00 0.105542D+00
1.800 0.820461D−01 0.860627D−01 0.900420D−01 0.939757D−01 0.978536D−01
1.900 0.768208D−01 0.804712D−01 0.840678D−01 0.876011D−01 0.910602D−01
2.000 0.721517D−01 0.754805D−01 0.787423D−01 0.819268D−01 0.850226D−01
2.100 0.679577D−01 0.710027D−01 0.739700D−01 0.768490D−01 0.796279D−01
2.200 0.641725D−01 0.669658D−01 0.696728D−01 0.722829D−01 0.747841D−01
2.300 0.607413D−01 0.633104D−01 0.657863D−01 0.681586D−01 0.704153D−01
2.400 0.576186D−01 0.599870D−01 0.622571D−01 0.644182D−01 0.664587D−01
2.500 0.547661D−01 0.569544D−01 0.590402D−01 0.610130D−01 0.628616D−01
2.600 0.521515D−01 0.541776D−01 0.560979D−01 0.579023D−01 0.595799D−01
2.700 0.497475D−01 0.516269D−01 0.533980D−01 0.550513D−01 0.565761D−01
2.800 0.475306D−01 0.492769D−01 0.509133D−01 0.524304D−01 0.538181D−01
2.900 0.454806D−01 0.471059D−01 0.486201D−01 0.500143D−01 0.512787D−01
3.000 0.435802D−01 0.450951D−01 0.464983D−01 0.477811D−01 0.489343D−01
3.100 0.418141D−01 0.432281D−01 0.445301D−01 0.457118D−01 0.467645D−01
3.200 0.401692D−01 0.414908D−01 0.427004D−01 0.437900D−01 0.447516D−01
3.300 0.386340D−01 0.398707D−01 0.409956D−01 0.420013D−01 0.428800D−01
3.400 0.371982D−01 0.383568D−01 0.394041D−01 0.403330D−01 0.411363D−01
3.500 0.358530D−01 0.369395D−01 0.379154D−01 0.387740D−01 0.395084D−01
3.600 0.345904D−01 0.356103D−01 0.365204D−01 0.373145D−01 0.379860D−01
3.700 0.334032D−01 0.343615D−01 0.352110D−01 0.359457D−01 0.365595D−01
3.800 0.322853D−01 0.331864D−01 0.339798D−01 0.346598D−01 0.352208D−01
3.900 0.312310D−01 0.320790D−01 0.328205D−01 0.334500D−01 0.339624D−01
4.000 0.302352D−01 0.310339D−01 0.317272D−01 0.323101D−01 0.327778D−01
4.100 0.292935D−01 0.300461D−01 0.306947D−01 0.312345D−01 0.316609D−01
4.200 0.284016D−01 0.291114D−01 0.297184D−01 0.302182D−01 0.306065D−01
4.300 0.275560D−01 0.282257D−01 0.287939D−01 0.292566D−01 0.296098D−01
4.400 0.267533D−01 0.273855D−01 0.279176D−01 0.283458D−01 0.286664D−01
4.500 0.259904D−01 0.265875D−01 0.270859D−01 0.274820D−01 0.277725D−01
4.600 0.252646D−01 0.258288D−01 0.262956D−01 0.266618D−01 0.269243D−01
4.700 0.245733D−01 0.251066D−01 0.255440D−01 0.258823D−01 0.261188D−01
4.800 0.239143D−01 0.244186D−01 0.248283D−01 0.251406D−01 0.253530D−01
4.900 0.232854D−01 0.237624D−01 0.241462D−01 0.244342D−01 0.246241D−01
5.000 0.226847D−01 0.231360D−01 0.234955D−01 0.237608D−01 0.239298D−01
t β
0.50 0.52 0.54 0.56 0.58
0.01 0.391772D−08
0.02 0.371680D−03 0.939964D−05 0.404234D−07
0.03 0.130495D−01 0.169605D−02 0.893449D−04 0.115097D−05 0.152735D−08
0.04 0.680714D−01 0.183678D−01 0.286174D−02 0.193321D−03 0.351487D−05
0.05 0.170007D+00 0.684450D−01 0.190661D−01 0.306418D−02 0.211287D−03
0.06 0.297583D+00 0.153336D+00 0.607468D−01 0.163477D−01 0.245053D−02
0.07 0.428249D+00 0.260102D+00 0.129875D+00 0.487872D−01 0.119843D−01
0.08 0.547759D+00 0.373510D+00 0.219075D+00 0.103472D+00 0.355059D−01
0.09 0.649618D+00 0.482197D+00 0.317871D+00 0.176950D+00 0.769517D−01
0.10 0.732249D+00 0.579576D+00 0.417048D+00 0.262290D+00 0.135746D+00
0.11 0.796660D+00 0.662765D+00 0.510177D+00 0.352173D+00 0.207879D+00
0.12 0.844973D+00 0.731240D+00 0.593536D+00 0.440582D+00 0.287906D+00
0.13 0.879627D+00 0.785795D+00 0.665456D+00 0.523313D+00 0.370513D+00
0.14 0.902978D+00 0.827855D+00 0.725645D+00 0.597843D+00 0.451359D+00
0.15 0.917137D+00 0.859059D+00 0.774628D+00 0.662958D+00 0.527319D+00
0.16 0.923911D+00 0.881041D+00 0.813371D+00 0.718366D+00 0.596414D+00
0.17 0.924812D+00 0.895307D+00 0.843025D+00 0.764370D+00 0.657596D+00
0.18 0.921085D+00 0.903197D+00 0.864775D+00 0.801619D+00 0.710505D+00
0.19 0.913744D+00 0.905866D+00 0.879753D+00 0.830941D+00 0.755259D+00
0.20 0.903612D+00 0.904299D+00 0.889000D+00 0.853224D+00 0.792275D+00
0.21 0.891351D+00 0.899321D+00 0.893441D+00 0.869353D+00 0.822150D+00
0.22 0.877495D+00 0.891623D+00 0.893884D+00 0.880163D+00 0.845564D+00
0.23 0.862470D+00 0.881775D+00 0.891029D+00 0.886419D+00 0.863221D+00
0.24 0.846619D+00 0.870251D+00 0.885473D+00 0.888811D+00 0.875812D+00
0.25 0.830215D+00 0.857438D+00 0.877724D+00 0.887948D+00 0.883987D+00
0.26 0.813475D+00 0.843656D+00 0.868212D+00 0.884361D+00 0.888347D+00
0.27 0.796572D+00 0.829167D+00 0.857297D+00 0.878513D+00 0.889435D+00
0.28 0.779642D+00 0.814183D+00 0.845282D+00 0.870802D+00 0.887738D+00
0.29 0.762792D+00 0.798880D+00 0.832421D+00 0.861571D+00 0.883684D+00
0.30 0.746107D+00 0.783398D+00 0.818926D+00 0.851113D+00 0.877653D+00
0.31 0.729651D+00 0.767851D+00 0.804972D+00 0.839675D+00 0.869973D+00
0.32 0.713473D+00 0.752334D+00 0.790705D+00 0.827470D+00 0.860931D+00
0.33 0.697611D+00 0.736918D+00 0.776247D+00 0.814677D+00 0.850774D+00
0.34 0.682091D+00 0.721663D+00 0.761697D+00 0.801446D+00 0.839717D+00
0.35 0.666934D+00 0.706617D+00 0.747138D+00 0.787906D+00 0.827944D+00
0.36 0.652151D+00 0.691813D+00 0.732636D+00 0.774163D+00 0.815611D+00
0.37 0.637750D+00 0.677282D+00 0.718246D+00 0.760308D+00 0.802854D+00
0.38 0.623734D+00 0.663043D+00 0.704012D+00 0.746414D+00 0.789787D+00
0.39 0.610105D+00 0.649111D+00 0.689970D+00 0.732546D+00 0.776508D+00
0.40 0.596858D+00 0.635497D+00 0.676147D+00 0.718753D+00 0.763100D+00
0.41 0.583990D+00 0.622208D+00 0.662566D+00 0.705078D+00 0.749633D+00
0.42 0.571495D+00 0.609246D+00 0.649243D+00 0.691557D+00 0.736166D+00
0.43 0.559365D+00 0.596614D+00 0.636190D+00 0.678217D+00 0.722748D+00
0.44 0.547593D+00 0.584310D+00 0.623416D+00 0.665081D+00 0.709421D+00
0.45 0.536169D+00 0.572330D+00 0.610928D+00 0.652166D+00 0.696218D+00
0.46 0.525084D+00 0.560672D+00 0.598728D+00 0.639487D+00 0.683169D+00
0.47 0.514328D+00 0.549330D+00 0.586818D+00 0.627053D+00 0.670296D+00
0.48 0.503893D+00 0.538298D+00 0.575198D+00 0.614873D+00 0.657618D+00
0.49 0.493767D+00 0.527569D+00 0.563865D+00 0.602951D+00 0.645149D+00
0.50 0.483941D+00 0.517138D+00 0.552817D+00 0.591291D+00 0.632902D+00
0.51 0.474406D+00 0.506995D+00 0.542050D+00 0.579894D+00 0.620885D+00
0.52 0.465152D+00 0.497134D+00 0.531560D+00 0.568760D+00 0.609105D+00
0.53 0.456169D+00 0.487547D+00 0.521342D+00 0.557887D+00 0.597566D+00
0.54 0.447448D+00 0.478227D+00 0.511390D+00 0.547275D+00 0.586270D+00
0.55 0.438980D+00 0.469165D+00 0.501699D+00 0.536919D+00 0.575220D+00
0.56 0.430756D+00 0.460354D+00 0.492261D+00 0.526816D+00 0.564416D+00
0.57 0.422769D+00 0.451786D+00 0.483072D+00 0.516963D+00 0.553855D+00
0.58 0.415009D+00 0.443454D+00 0.474125D+00 0.507354D+00 0.543538D+00
0.59 0.407468D+00 0.435350D+00 0.465413D+00 0.497985D+00 0.533461D+00
0.60 0.400140D+00 0.427467D+00 0.456929D+00 0.488851D+00 0.523621D+00
0.61 0.393016D+00 0.419799D+00 0.448669D+00 0.479947D+00 0.514015D+00
0.62 0.386090D+00 0.412337D+00 0.440625D+00 0.471267D+00 0.504639D+00
0.63 0.379354D+00 0.405076D+00 0.432791D+00 0.462805D+00 0.495489D+00
0.64 0.372803D+00 0.398009D+00 0.425160D+00 0.454556D+00 0.486560D+00
0.65 0.366428D+00 0.391130D+00 0.417728D+00 0.446516D+00 0.477847D+00
0.66 0.360225D+00 0.384432D+00 0.410487D+00 0.438677D+00 0.469347D+00
0.67 0.354187D+00 0.377910D+00 0.403433D+00 0.431035D+00 0.461053D+00
0.68 0.348309D+00 0.371557D+00 0.396559D+00 0.423585D+00 0.452962D+00
0.69 0.342585D+00 0.365369D+00 0.389860D+00 0.416320D+00 0.445068D+00
0.70 0.337010D+00 0.359341D+00 0.383331D+00 0.409237D+00 0.437367D+00
0.71 0.331579D+00 0.353466D+00 0.376966D+00 0.402328D+00 0.429852D+00
0.72 0.326287D+00 0.347740D+00 0.370761D+00 0.395591D+00 0.422521D+00
0.73 0.321129D+00 0.342157D+00 0.364710D+00 0.389019D+00 0.415367D+00
0.74 0.316101D+00 0.336715D+00 0.358809D+00 0.382609D+00 0.408386D+00
0.75 0.311199D+00 0.331407D+00 0.353053D+00 0.376354D+00 0.401573D+00
0.76 0.306418D+00 0.326230D+00 0.347438D+00 0.370252D+00 0.394924D+00
0.77 0.301755D+00 0.321180D+00 0.341959D+00 0.364296D+00 0.388435D+00
0.78 0.297205D+00 0.316252D+00 0.336613D+00 0.358484D+00 0.382100D+00
0.79 0.292765D+00 0.311443D+00 0.331395D+00 0.352810D+00 0.375916D+00
0.80 0.288432D+00 0.306748D+00 0.326301D+00 0.347271D+00 0.369878D+00
0.81 0.284201D+00 0.302165D+00 0.321327D+00 0.341863D+00 0.363982D+00
0.82 0.280071D+00 0.297690D+00 0.316471D+00 0.336581D+00 0.358224D+00
0.83 0.276036D+00 0.293319D+00 0.311728D+00 0.331423D+00 0.352601D+00
0.84 0.272096D+00 0.289050D+00 0.307094D+00 0.326385D+00 0.347108D+00
0.85 0.268246D+00 0.284879D+00 0.302568D+00 0.321462D+00 0.341742D+00
0.86 0.264484D+00 0.280803D+00 0.298145D+00 0.316653D+0Ó 0.336500D+00
0.87 0.260807D+00 0.276820D+00 0.293823D+00 0.311953D+00 0.331377D+00
0.88 0.257212D+00 0.272926D+00 0.289598D+00 0.307359D+00 0.326370D+00
0.89 0.253698D+00 0.269119D+00 0.285467D+00 0.302869D+00 0.321477D+00
0.90 0.250261D+00 0.265397D+00 0.281429D+00 0.298479D+00 0.316693D+00
0.91 0.246900D+00 0.261756D+00 0.277480D+00 0.294187D+00 0.312017D+00
0.92 0.243612D+00 0.258195D+00 0.273617D+00 0.289989D+00 0.307444D+00
0.93 0.240395D+00 0.254711D+00 0.269838D+00 0.285883D+00 0.302972D+00
0.94 0.237246D+00 0.251302D+00 0.266142D+00 0.281867D+00 0.298598D+00
0.95 0.234165D+00 0.247966D+00 0.262524D+00 0.277937D+00 0.294320D+00
0.96 0.231148D+00 0.244701D+00 0.258984D+00 0.274092D+00 0.290135D+00
0.97 0.228195D+00 0.241504D+00 0.255519D+00 0.270329D+00 0.286040D+00
0.98 0.225303D+00 0.238374D+00 0.252127D+00 0.266646D+00 0.282032D+00
0.99 0.222470D+00 0.235309D+00 0.248805D+00 0.263040D+00 0.278110D+00
1.00 0.219696D+00 0.232307D+00 0.245553D+00 0.259510D+00 0.274271D+00
1.05 0.206637D+00 0.218184D+00 0.230259D+00 0.242923D+00 0.256245D+00
1.10 0.194806D+00 0.205400D+00 0.216430D+00 0.227940D+00 0.239984D+00
1.15 0.184050D+00 0.193788D+00 0.203881D+00 0.214362D+00 0.225269D+00
1.20 0.174239D+00 0.183206D+00 0.192458D+00 0.202019D+00 0.211912D+00
1.25 0.165261D+00 0.173533D+00 0.182029D+00 0.190764D+00 0.199752D+00
1.30 0.157022D+00 0.164666D+00 0.172480D+00 0.180473D+00 0.188651D+00
1.35 0.149441D+00 0.156514D+00 0.163712D+00 0.171037D+00 0.178488D+00
1.40 0.142446D+00 0.149002D+00 0.155642D+00 0.162363D+00 0.169161D+00
1.45 0.135977D+00 0.142061D+00 0.148195D+00 0.154371D+00 0.160580D+00
1.50 0.129980D+00 0.135634D+00 0.141307D+00 0.146989D+00 0.152667D+00
1.60 0.119222D+00 0.124123D+00 0.128993D+00 0.133818D+00 0.138579D+00
1.70 0.109864D+00 0.114131D+00 0.1Ì8329D+00 0.122442D+00 0.126447D+00
1.80 0.101664D+00 0.105393D+00 0.109026D+00 0.112542D+00 0.115921D+00
1.90 0.944323D−01 0.977029D−01 0.100855D+00 0.103869D+00 0.106724D+00
2.00 0.880163D−01 0.908931D−01 0.936357D−01 0.962246D−01 0.986376D−01
2.10 0.822933D−01 0.848301D−01 0.872213D−01 0.894477D−01 0.914877D−01
2.20 0.771631D−01 0.794049D−01 0.814931D−01 0.834090D−01 0.851319D−01
2.30 0.725433D−01 0.745282D−01 0.763538D−01 0.780025D−01 0.794545D−01
2.40 0.683658D−01 0.701257D−01 0.717229D−01 0.731406D−01 0.743602D−01
2.50 0.645737D−01 0.661358D−01 0.675334D−01 0.687506D−01 0.697701D−01
2.60 0.611190D−01 0.625067D−01 0.637292D−01 0.647718D−01 0.656182D−01
2.70 0.579611D−01 0.591944D−01 0.602630D−01 0.611529D−01 0.618492D−01
2.80 0.550658D−01 0.561619D−01 0.570945D−01 0.578505D−01 0.584163D−01
2.90 0.524033D−01 0.533774D−01 0.541895D−01 0.548278D−01 0.552798D−01
3.00 0.499484D−01 0.508134D−01 0.515186D−01 0.520532D−01 0.524055D−01
3.10 0.476792D−01 0.484464D−01 0.490565D−01 0.494993D−01 0.497643D−01
3.20 0.455765D−01 0.462561D−01 0.467812D−01 0.471427D−01 0.473309D−01
3.30 0.436238D−01 0.442244D−01 0.446736D−01 0.449629D−01 0.450837D−01
3.40 0.418064D−01 0.423360D−01 0.427171D−01 0.429422D−01 0.430035D−01
3.50 0.401118D−01 0.405770D−01 0.408971D−01 0.410650D−01 0.410737D−01
3.60 0.385285D−01 0.389355D−01 0.392007D−01 0.393175D−01 0.392799D−01
3.70 0.370466D−01 0.374009D−01 0.376165D−01 0.376877D−01 0.376090D−01
3.80 0.356572D−01 0.359636D−01 0.361345D−01 0.361650D−01 0.360499D−01
3.90 0.343525D−01 0.346153D−01 0.347459D−01 0.347397D−01 0.345925D−01
4.00 0.331254D−01 0.333485D−01 0.334426D−01 0.334037D−01 0.332279D−01
4.10 0.319697D−01 0.321565D−01 0.322175D−01 0.321492D−01 0.319482D−01
4.20 0.308795D−01 0.310333D−01 0.310643D−01 0.309695D−01 0.307462D−01
4.30 0.298499D−01 0.299734D−01 0.299773D−01 0.298587D−01 0.296155D−01
4.40 0.288762D−01 0.289720D−01 0.289512D−01 0.288114D−01 0.285506D−01
4.50 0.279543D−01 0.280248D−01 0.279815D−01 0.278225D−01 0.275462D−01
4.60 0.270804D−01 0.271276D−01 0.270639D−01 0.268876D−01 0.265977D−01
4.70 0.262511D−01 0.262770D−01 0.261947D−01 0.260029D−01 0.257009D−01
4.80 0.254632D−01 0.254695D−01 0.253703D−01 0.251646D−01 0.248519D−01
4.90 0.247140D−01 0.247022D−01 0.245876D−01 0.243694D−01 0.240474D−01
5.00 0.240008D−01 0.239724D−01 0.238438D−01 0.236144D−01 0.232842D−01
t β
0.60 0.62 0.64 0.66 0.68
0.04 0.753752D−08
0.05 0.378924D−05 0.731813D−08
0.06 0.147234D−03 0.200982D−05 0.223739D−08
0.07 0.153136D−02 0.690699D−04 0.553632D−06
0.08 0.749966D−02 0.739405D−03 0.210256D−04 0.716141D−07
0.09 0.230965D−01 0.390108D−02 0.261311D−03 0.370366D−05 0.337870D−08
0.10 0.525756D−01 0.130560D−01 0.160026D−02 0.609465D−04 0.304556D−06
0.11 0.974210D−01 0.321607D−01 0.612268D−02 0.474054D−03 0.786366D−05
0.12 0.156134D+00 0.639525D−01 0.169288D−01 0.221323D−02 0.878290D−04
0.13 0.225217D+00 0.109008D+00 0.371360D−01 0.721405D−02 0.548432D−03
0.14 0.300398D+00 0.165831D+00 0.687774D−01 0.181315D−01 0.226080D−02
0.15 0.377571D+00 0.231546D+00 0.112233D+00 0.375915D−01 0.687742D−02
0.16 0.453305D+00 0.302712D+00 0.166304D+00 0.673907D−01 0.166711D−01
0.17 0.525031D+00 0.375960D+00 0.228682D+00 0.108041D+00 0.339983D−01
0.18 0.591020D+00 0.448380D+00 0.296526D+00 0.158755D+00 0.606643D−01
0.19 0.650257D+00 0.517691D+00 0.366947D+00 0.217746D+00 0.974831D−01
0.20 0.702287D+00 0.582258D+00 0.437343D+00 0.282650D+00 0.144155D+00
0.21 0.747063D+00 0.641029D+00 0.505567D+00 0.350926D+00 0.199419D+00
0.22 0.784823D+00 0.693432D+00 0.569988D+00 0.420159D+00 0.261354D+00
0.23 0.815983D+00 0.739269D+00 0.629467D+00 0.488254D+00 0.327720D+00
0.24 0.841061D+00 0.778617D+00 0.683296D+00 0.553518D+00 0.396241D+00
0.25 0.860627D+00 0.811740D+00 0.731122D+00 0.614686D+00 0.464822D+00
0.26 0.875259D+00 0.839022D+00 0.772863D+00 0.670886D+00 0.531670D+00
0.27 0.885522D+00 0.860919D+00 0.808641D+00 0.721591D+00 0.595351D+00
0.28 0.891949D+00 0.877921D+00 0.838719D+00 0.766554D+00 0.654798D+00
0.29 0.895035D+00 0.890521D+00 0.863450D+00 0.805748D+00 0.709280D+00
0.30 0.895233D+00 0.899201D+00 0.883243D+00 0.839311D+00 0.758360D+00
0.31 0.892950D+00 0.904420D+00 0.898532D+00 0.867494D+00 0.801842D+00
0.32 0.888549D+00 0.906603D+00 0.909753D+00 0.890626D+00 0.839719D+00
0.33 0.882355D+00 0.906144D+00 0.917336D+00 0.909082D+00 0.872131D+00
0.34 0.874651D+00 0.903399D+00 0.921689D+00 0.923258D+00 0.899317D+00
0.35 0.865688D+00 0.898690D+00 0.923194D+00 0.933558D+00 0.921589D+00
0.36 0.855684D+00 0.892304D+00 0.922207D+00 0.940375D+00 0.939301D+00
0.37 0.844830D+00 0.884496D+00 0.919051D+00 0.944087D+00 0.952826D+00
0.38 0.833291D+00 0.875493D+00 0.914022D+00 0.945052D+00 0.962548D+00
0.39 0.821211D+00 0.865495D+00 0.907384D+00 0.943600D+00 0.968840D+00
0.40 0.808712D+00 0.854677D+00 0.899374D+00 0.940037D+00 0.972067D+00
0.41 0.795902D+00 0.843192D+00 0.890205D+00 0.934642D+00 0.972571D+00
0.42 0.782871D+00 0.831174D+00 0.880064D+00 0.927667D+00 0.970673D+00
0.43 0.769698D+00 0.818739D+00 0.869116D+00 0.919340D+00 0.966670D+00
0.44 0.756450D+00 0.805989D+00 0.857510D+00 0.909865D+00 0.960834D+00
0.45 0.743183D+00 0.793011D+00 0.845372D+00 0.899424D+00 0.953415D+00
0.46 0.729946D+00 0.779880D+00 0.832815D+00 0.888178D+00 0.944635D+00
0.47 0.716780D+00 0.766662D+00 0.819938D+00 0.876270D+00 0.934698D+00
0.48 0.703717D+00 0.753412D+00 0.806826D+00 0.863827D+00 0.923784D+00
0.49 0.690788D+00 0.740178D+00 0.793553D+00 0.850960D+00 0.912054D+00
0.50 0.678016D+00 0.727000D+00 0.780183D+00 0.837766D+00 0.899653D+00
0.51 0.665419D+00 0.713913D+00 0.766773D+00 0.824331D+00 0.886707D+00
0.52 0.653015D+00 0.700945D+00 0.753369D+00 0.810729D+00 0.873328D+00
0.53 0.640814D+00 0.688120D+00 0.740012D+00 0.797024D+00 0.859616D+00
0.54 0.628828D+00 0.675460D+00 0.726738D+00 0.783273D+00 0.845656D+00
0.55 0.617063D+00 0.662979D+00 0.713575D+00 0.769524D+00 0.831526D+00
0.56 0.605526D+00 0.650692D+00 0.700549D+00 0.755819D+00 0.817290D+00
0.57 0.594219D+00 0.638610D+00 0.687681D+00 0.742193D+00 0.803007D+00
0.58 0.583146D+00 0.626740D+00 0.674987D+00 0.728676D+00 0.788727D+00
0.59 0.572308D+00 0.615090D+00 0.662482D+00 0.715295D+00 0.774491D+00
0.60 0.561704D+00 0.603663D+00 0.650177D+00 0.702071D+00 0.760337D+00
0.61 0.551334D+00 0.592463D+00 0.638082D+00 0.689022D+00 0.746296D+00
0.62 0.541197D+00 0.581493D+00 0.626204D+00 0.676163D+00 0.732396D+00
0.63 0.531289D+00 0.570751D+00 0.614547D+00 0.663506D+00 0.718658D+00
0.64 0.521608D+00 0.560240D+00 0.603116D+00 0.651060D+00 0.705101D+00
0.65 0.512152D+00 0.549957D+00 0.591913D+00 0.638834D+00 0.691742D+00
0.66 0.502916D+00 0.539901D+00 0.580940D+00 0.626834D+00 0.678592D+00
0.67 0.493897D+00 0.530069D+00 0.570196D+00 0.615063D+00 0.665663D+00
0.68 0.485090D+00 0.520460D+00 0.559682D+00 0.603524D+00 0.652962D+00
0.69 0.476492D+00 0.511069D+00 0.549396D+00 0.592220D+00 0.640495D+00
0.70 0.468098D+00 0.501895D+00 0.539336D+00 0.581150D+00 0.628268D+00
0.71 0.459903D+00 0.492932D+00 0.529499D+00 0.570314D+00 0.616282D+00
0.72 0.451904D+00 0.484177D+00 0.519884D+00 0.559713D+00 0.604541D+00
0.73 0.444095D+00 0.475626D+00 0.510487D+00 0.549343D+00 0.593044D+00
0.74 0.436472D+00 0.467275D+00 0.501304D+00 0.539202D+00 0.581792D+00
0.75. 0.429030D+00 0.459120D+00 0.492333D+00 0.529289D+00 0.570783D+00
0.76 0.421766D+00 0.451156D+00 0.483568D+00 0.519600D+00 0.560017D+00
0.77 0.414674D+00 0.443379D+00 0.475006D+00 0.510131D+00 0.549490D+00
0.78 0.407749D+00 0.435784D+00 0.466643D+00 0.500880D+00 0.539200D+00
0.79 0.400989D+00 0.428368D+00 0.458475D+00 0.491841D+00 0.529145D+00
0.80 0.394387D+00 0.421125D+00 0.450497D+00 0.483012D+00 0.519319D+00
0.81 0.387941D+00 0.414052D+00 0.442705D+00 0.474387D+00 0.509721D+00
0.82 0.381645D+00 0.407145D+00 0.435095D+00 0.465964D+00 0.500345D+00
0.83 0.375497D+00 0.400398D+00 0.427663D+00 0.457737D+00 0.491189D+00
0.84 0.369491D+00 0.393809D+00 0.420403D+00 0.449702D+00 0.482246D+00
0.85 0.363624D+00 0.387372D+00 0.413313D+00 0.441854D+00 0.473513D+00
0.86 0.357892D+00 0.381084D+00 0.406387D+00 0.434191D+00 0.464986D+00
0.87 0.352292D+00 0.374941D+00 0.399623D+00 0.426706D+.00 0.456660D+00
0.88 0.346820D+00 0.368940D+00 0.393014D+00 0.419396D+00 0.448530D+00
0.89 0.341472D+00 0.363076D+00 0.386558D+00 0.412256D+00 0.440592D+00
0.90 0.336245D+00 0.357345D+00 0.380251D+00 0.405283D+00 0.432842D+00
0.91 0.331136D+00 0.351744D+00 0.374089D+00 0.398472D+00 0.425274D+00
0.92 0.326141D+00 0.346271D+00 0.368067D+00 0.391818D+00 0.417884D+00
0.93 0.321257D+00 0.340920D+00 0.362183D+00 0.385319D+00 0.410668D+00
0.94 0.316481D+00 0.335689D+00 0.356432D+00 0.378969D+00 0.403621D+00
0.95 0.311811D+00 0.330575D+00 0.350811D+00 0.372765D+00 0.396740D+00
0.96 0.307243D+00 0.325574D+00 0.345317D+00 0.366703D+00 0.390020D+00
0.97 0.302775D+00 0.320684D+00 0.339946D+00 0.360780D+00 0.383456D+00
0.98 0.298404D+00 0.315902D+00 0.334695D+00 0.354992D+00 0.377044D+00
0.99 0.294127D+00 0.311224D+00 0.329561D+00 0.349334D+00 0.370782D+00
1.00 0.289941D+00 0.306647D+00 0.324541D+00 0.343805D+00 0.364664D+00
1.05 0.270305D+00 0.285199D+00 0.301039D+00 0.317957D+00 0.336112D+00
1.10 0.252620D+00 0.265918D+00 0.279956D+00 0.294828D+00 0.310641D+00
1.15 0.236643D+00 0.248533D+00 0.260990D+00 0.274075D+00 0.287857D+00
1.20 0.222166D+00 0.232811D+00 0.243879D+00 0.255404D+00 0.267422D+00
1.25 0.209010D+00 0.218553D+00 0.228397D+00 0.238556D+00 0.249042D+00
1.30 0.197020D+00 0.205586D+00 0.214351D+00 0.223312D+00 0.232463D+00
1.35 0.186065D+00 0.193762D+00 0.201572D+00 0.209481D+00 0.217468D+00
1.40 0.176028D+00 0.182951D+00 0.189916D+00 0.196898D+00 0.203867D+00
1.45 0.166810D+00 0.173043D+00 0.179257D+00 0.185422D+00 0.191499D+00
1.50 0.158324D+00 0.163939D+00 0.169486D+00 0.174928D+00 0.180222D+00
1.60 0.143256D+00 0.147820D+00 0.152240D+00 0.156473D+00 0.160471D+00
1.70 0.130322D+00 0.134036D+00 0.137553D+00 0.140831D+00 0.143817D+00
1.80 0.119135D+00 0.122155D+00 0.124945D+00 0.127461D+00 0.129653D+00
1.90 0.109391D+00 0.111842D+00 0.114041D+00 0.115946D+00 0.117510D+00
2.00 0.100849D+00 0.102830D+00 0.104546D+00 0.105958D+00 0.107023D+00
2.10 0.933166D−01 0.949066D−01 0.962261D−01 0.972392D−01 0.979051D−01
2.20 0.866384D−01 0.879024D−01 0.888946D−01 0.895821D−01 0.899282D−01
2.30 0.806879D−01 0.816785D−01 0.823994D−01 0.828207D−01 0.829094D−01
2.40 0.753614D−01 0.761218D−01 0.766169D−01 0.768197D−01 0.767010D−01
2.50 0.705730D−01 0.711388D−01 0.714454D−01 0.714686D−01 0.711823D−01
2.60 0.662512D−01 0.666521D−01 0.668009D−01 0.666760D−01 0.662545D−01
2.70 0.623362D−01 0.625969D−01 0.626132D−01 0.623662D−01 0.618356D−01
2.80 0.587775D−01 0.589186D−01 0.588236D−01 0.584758D−01 0.578575D−01
2.90 0.555321D−01 0.555711D−01 0.553825D−01 0.549515D−01 0.542629D−01
3.00 0.525636D−01 0.525153D−01 0.522479D−01 0.517484D−01 0.510037D−01
3.10 0.49.8407D−01 0.497176D−01 0.493837D−01 0.488280D−01 0.480391D−01
3.20 0.473363D−01 0.471491D−01 0.467594D−01 0.461576D−01 0.453343D−01
3.30 0.450273D−01 0.447850D−01 0.443484D−01 0.437092D−01 0.428594D−01
3.40 0.428932D−01 0.426038D−01 0.421279D−01 0.414584D−01 0.405889D−01
3.50 0.409165D−01 0.405867D−01 0.400779D−01 0.393843D−01 0.385005D−01
3.60 0.390817D−01 0.387172D−01 0.381811D−01 0.374685D−01 0.365752D−01
3.70 0.373751D−01 0.369811D−01 0.364224D−01 0.356951D−01 0.347961D−01
3.80 0.357849D−01 0.353656D−01 0.347884D−01 0.340502D−01 0.331486D−01
3.90 0.343003D−01 0.338596D−01 0.332673D−01 0.325213D−01 0.316199D−01
4.00 0.329120D−01 0.324531D−01 0.318489D−01 0.310977D−01 0.301986D−01
4.10 0.316117D−01 0.311375D−01 0.305239D−01 0.297697D−01 0.288748D−01
4.20 0.303919D−01 0.299049D−01 0.292841D−01 0.285289D−01 0.276397D−01
4.30 0.292458D−01 0.287482D−01 0.281222D−01 0.273676D−01 00.264854D−01
4.40 0.281675D−01 0.276613D−01 0.270316D−01 0.262791D−01 0.254048D−01
4.50 0.271517D−01 0.266384D−01 0.260066D−01 0.252572D−01 0.243918D−01
4.60 0.261933D−01 0.256746D−01 0.250419D−01 0.242966D−01 0.234407D−01
4.70 0.252882D−01 0.247652D−01 0.241327D−01 0.233924D−01 0.225465D−01
4.80 0.244323D−01 0.239062D−01 0.232748D−01 0.225401D−01 0.217046D−01
4.90 0.236219D−01 0.230937D−01 0.224643D−01 0.217359D−01 0.209111D−01
5.00 0.228539D−01 0.223245D−01 0.216977D−01 0.209759D−01 0.201622D−01
t β
0.70 0.72 0.74 0.76 0.78
0.11 0.802539D−08
0.12 0.416131D−06
0.13 0.801084D−05 0.535037D−08
0.14 0.768242D−04 0.237760D−06
0.15 0.445589D−03 0.437149D−05 0.107936D−08
0.16 0.178343D−02 0.423297D−04 0.555972D−07
0.17 0.540032D−02 0.254785D−03 0.120269D−05
0.18 0.132108D−01 0.107115D−02 0.136544D−04 0.410099D−08
0.19 0.273863D−01 0.342000D−02 0.952427D−04 0.132391D−06
0.20 0.498423D−01 0.881655D−02 0.457233D−03 0.211103D−05
0.21 0.817871D−01 0.192061D−01 0.164189D−02 0.196205D−04 0.353918D−08
0.22 0.123485D+00 0.365905D−01 0.469218D−02 0.120010D−03 0.103722D−06
0.23 0.174261D+00 0.625878D−01 0.111829D−01 0.528740D−03 0.158778D−05
0.24 0.232686D+00 0.980918D−01 0.230376D−01 0.179620D−02 0.146465D−04
0.25 0.296847D+00 0.143121D+00 0.421740D−01 0.495599D−02 0.906410D−04
0.26 0.364621D+00 0.196860D+00 0.701107D−01 0.115610D−01 0.408322D−03
0.27 0.433905D+00 0.257841D+00 0.107666D+00 0.235257D−01 0.142562D−02
0.28 0.502783D+00 0.324180D+00 0.154827D+00 0.428024D−01 0.405009D−02
0.29 0.569621D+00 0.393822D+00 0.210783D+00 0.710024D−01 0.972639D−02
0.30 0.633115D+00 0.464738D+00 0.274087D+00 0.109084D+00 0.203496D−01
0.31 0.692288D+00 0.535079D+00 0.342878D+00 0.157188D+00 0.379935D−01
0.32 0.746468D+00 0.603260D+00 0.415108D+00 0.214639D+00 0.645330D−01
0.33 0.795254D+00 0.668005D+00 0.488734D+00 0.280076D+00 0.101284D+00
0.34 0.838463D+00 0.728348D+00 0.561872D+00 0.351658D+00 0.148762D+00
0.35 0.876092D+00 0.783619D+00 0.632888D+00 0.427292D+00 0.206607D+00
0.36 0.908274D+00 0.833406D+00 0.700445D+00 0.504840D+00 0.273657D+00
0.37 0.935238D+00 0.877515D+00 0.763518D+00 0.582284D+00 0.348137D+00
0.38 0.957285D+00 0.915930D+00 0.821377D+00 0.657835D+00 0.427889D+00
0.39 0.974756D+00 0.948772D+00 0.873563D+00 0.730005D+00 0.510608D+00
0.40 0.988018D+00 0.976262D+00 0.919839D+00 0.797622D+00 0.594035D+00
0.41 0.997443D+00 0.998694D+00 0.960161D+00 0.859834D+00 0.676112D+00
0.42 0.100340D+01 0.101641D+01 0.994629D+00 0.916075D+00 0.755071D+00
0.43 0.100625D+01 0.102977D+01 0.102346D+01 0.966034D+00 0.829487D+00
0.44 0.100634D+01 0.103916D+01 0.104694D+01 0.100961D+01 0.898284D+00
0.45 0.100398D+01 0.104496D+01 0.106542D+01 0.104688D+01 0.960716D+00
0.46 0.999474D+00 0.104752D+01 0.107928D+01 0.107803D+01 0.101634D+01
0.47 0.993096D+00 0.104720D+01 0.108891D+01 0.110337D+01 0.106495D+01
0.48 0.985095D+00 0.104434D+01 0.109471D+01 0.112326D+01 0.110657D+01
0.49 0.975701D+00 0.103923D+01 0.109706D+01 0.113809D+01 0.114138D+01
0.50 0.965119D+00 0.103218D+01 0.109634D+01 0.114829D+01 0.116967D+01
0.51 0.953534D+00 0.102342D+01 0.109290D+01 0.115430D+01 0.119182D+01
0.52 0.941112D+00 0.101322D+01 0.108707D+01 0.115653D+01 0.120828D+01
0.53 0.927999D+00 0.100177D+01 0.107915D+01 0.115539D+01 0.121951D+01
0.54 0.914327D+00 0.989279D+00 0.106943D+01 0.115127D+01 0.122600D+01
0.55 0.900212D+00 0.975915D+00 0.105816D+01 0.114454D+01 0.122823D+01
0.56 0.885755D+00 0.961835D+00 0.104558D+01 0.113554D+01 0.122666D+01
0.57 0.871047D+00 0.947176D+00 0.103189D+01 0.112457D+01 0.122174D+01
0.58 0.856167D+00 0.932061D+00 0.101728D+01 0.111192D+01 0.121388D+01
0.59 0.841183D+00 0.916597D+00 0.100192D+01 0.109785D+01 0.120348D+01
0.60 0.826154D+00 0.900879D+00 0.985952D+00 0.108258D+01 0.119089D+01
0.61 0.811133D+00 0.884991D+00 0.969517D+00 0.106634D+01 0.117643D+01
0.62 0.796165D+00 0.869005D+00 0.952726D+00 0.104930D+01 0.116041D+01
0.63 0.781287D+00 0.852983D+00 0.935682D+00 0.103163D+01 0.114309D+01
0.64 0.766533D+00 0.836982D+00 0.918475D+00 0.101347D+01 0.112470D+01
0.65 0.751931D+00 0.821048D+00 0.901183D+00 0.994947D+00 0.110545D+01
0.66 0.737504D+00 0.805221D+00 0.883872D+00 0.976182D+00 0.108554D+01
0.67 0.723272D+00 0.789536D+00 0.866603D+00 0.957269D+00 0.106513D+01
0.68 0.709251D+00 0.774023D+00 0.849425D+00 0.938292D+00 0.104436D+01
0.69 0.695454D+00 0.758706D+00 0.832381D+00 0.919326D+00 0.102337D+01
0.70 0.681893D+00 0.743606D+00 0.815508D+00 0.900434D+00 0.100225D+01
0.71 0.668576D+00 0.728739D+00 0.798836D+00 0.881670D+00 0.981101D+00
0.72 0.655509D+00 0.714120D+00 0.782393D+00 0.863080D+00 0.960011D+00
0.73 0.642696D+00 0.699759D+00 0.766198D+00 0.844704D+00 0.939045D+00
0.74 0.630141D+00 0.685666D+00 0.750270D+00 0.826575D+00 0.918264D+00
0.75 0.617847D+00 0.671846D+00 0.734623D+00 0.808718D+00 0.897716D+00
0.76 0.605812D+00 0.658304D+00 0.719268D+00 0.791158D+00 0.877445D+00
0.77 0.594038D+00 0.645043D+00 0.704213D+00 0.773910D+00 0.857483D+00
0.78 0.582524D+00 0.632065D+00 0.689465D+00 0.756991D+00 0.837860D+00
0.79 0.571266D+00 0.619370D+00 0.675027D+00 0.740409D+00 0.818599D+00
0.80 0.560263D+00 0.606957D+00 0.660903D+00 0.724174D+00 0.799716D+00
0.81 0.549513D+00 0.594826D+00 0.647093D+00 0.708292D+00 0.781226D+00
0.82 0.539010D+00 0.582973D+00 0.633597D+00 0.692764D+00 0.763138D+00
0.83 0.528753D+00 0.571395D+00 0.620414D+00 0.677594D+00 0.745461D+00
0.84 0.518736D+00 0.560090D+00 0.607541D+00 0.662780D+00 0.728197D+00
0.85 0.508956D+00 0.549054D+00 0.594976D+00 0.648322D+00 0.711349D+00
0.86 0.499407D+00 0.538281D+00 0.582714D+00 0.634217D+00 0.694917D+00
0.87 0.490086D+00 0.527768D+00 0.570751D+00 0.620462D+00 0.678898D+00
0.88 0.480987D+00 0.517509D+00 0.559083D+00 0.607051D+00 0.663290D+00
0.89 0.472106D+00 0.507500D+00 0.547704D+00 0.593981D+00 0.648089D+00
0.90 0.463438D+00 0.497735D+00 0.536609D+00 0.581245D+00 0.633288D+00
0.91 0.454978D+00 0.488210D+00 0.525793D+00 0.568838D+00 0.618882D+00
0.92 0.446721D+00 0.478918D+00 0.515248D+00 0.556752D+00 0.604863D+00
0.93 0.438662D+00 0.469854D+00 0.504970D+00 0.544982D+00 0.591224D+00
0.94 0.430796D+00 0.461013D+00 0.494952D+00 0.533521D+00 0.577958D+00
0.95 0.423119D+00 0.452390D+00 0.485189D+00 0.522361D+00 0.565056D+00
0.96 0.415625D+00 0.443978D+00 0.475673D+00 0.511495D+00 0.552509D+00
0.97 0.408311D+00 0.435774D+00 0.466399D+00 0.500916D+00 0.540308D+00
0.98 0.401170D+00 0.427770D+00 0.457360D+00 0.490617D+00 0.528446D+00
0.99 0.394200D+00 0.419963D+00 0.448551D+00 0.480590D+00 0.516914D+00
1.00 0.387395D+00 0.412347D+00 0.439966D+00 0.470828D+00 0.505701D+00
1.05 0.355701D+00 0.376959D+00 0.400185D+00 0.425753D+00 0.454141D+00
1.10 0.327524D+00 0.345630D+00 0.365142D+00 0.386282D+00 0.409320D+00
1.15 0.302412D+00 0.317828D+00 0.334203D+00 0.351647D+00 0.370279D+00
1.20 0.279972D+00 0.293092D+00 0.306816D+00 0.321173D+00 0.336177D+00
1.25 0.259863D+00 0.271019D+00 0.282501D+00 0.294279D+00 0.306293D+00
1.30 0.241790D+00 0.251266D+00 0.260848D+00 0.270466D+00 0.280013D+00
1.35 0.225501D+00 0.233536D+00 0.241504D+00 0.249312D+00 0.256820D+00
1.40 0.210779D+00 0.217574D+00 0.224170D+00 0.230456D+00 0.236276D+00
1.45 0.197435D+00 0.203163D+00 0.208590D+00 0.213595D+00 0.218014D+00
1.50 0.185309D+00 0.190115D+00 0.194544D+00 0.198469D+00 0.201723D+00
1.60 0.164168D+00 0.167488D+00 0.170330D+00 0.172569D+00 0.174046D+00
1.70 0.146448D+00 0.148647D+00 0.150320D+00 0.151348D+00 0.151590D+00
1.80 0.131460D+00 0.132810D+00 0.133615D+00 0.133771D+00 0.133151D+00
1.90 0.118676D+00 0.119378D+00 0.119538D+00 0.119063D+00 0.117846D+00
2.00 0.107688D+00 0.107895D+00 0.107573D+00 0.106644D+00 0.105016D+00
2.10 0.981775D−01 0.980035D−01 0.973231D−01 0.960682D−01 0.941615D−01
2.20 0.898914D−01 0.894253D−01 0.884781D−01 0.869920D−01 0.849027D−01
2.30 0.826289D−01 0.819389D−01 0.807947D−01 0.791477D−01 0.769447D−01
2.40 0.762284D−01 0.753671D−01 0.740791D−01 0.723236D−01 0.700569D−01
2.50 0.705586D−01 0.695672D−01 0.681760D−01 0.663512D−01 0.640573D−01
2.60 0.655121D−01 0.644229D−01 0.629600D−01 0.610953D−01 0.588004D−01
2.70 0.610005D−01 0.598390D−01 0.583285D−01 0.564461D−01 0.541690D−01
2.80 0.569507D−01 0.557369D−01 0.541974D−01 0.523138D−01 0.500683D−01
2.90 0.533013D−01 0.520511D−01 0.504972D−01 0.486248D−01 0.464202D−01
3.00 0.500009D−01 0.487271D−01 0.471699D−01 0.453179D−01 0.431608D−01
3.10 0.470062D−01 0.457187D−01 0.441669D−01 0.423421D−01 0.402368D−01
3.20 0.442803D−01 0.429871D−01 0.414474D−01 0.396545D−01 0.376038D−01
3.30 0.417916D−01 0.404992D−01 0.389765D−01 0.372191D−01 0.352243D−01
3.40 0.395133D−01 0.382266D−01 0.367248D−01 0.350052D−01 0.330669D−01
3.50 0.374220D−01 0.361450D−01 0.346670D−01 0.329867D−01 0.311046D−01
3.60 0.354977D−01 0.342335D−01 0.327812D−01 0.311411D−01 0.293147D−01
3.70 0.337228D−01 0.324738D−01 0.310489D−01 0.294492D−01 0.276774D−01
3.80 0.320821D−01 0.308502D−01 0.294536D−01 0.278943D−01 0.261758D−01
3.90 0.305623D−01 0.293489D−01 0.279812D−01 0.264619D−01 0.247953D−01
4.00 0.291516D−01 0.279578D−01 0.266192D−01 0.251394D−01 0.235232D−01
4.10 0.278398D−01 0.266662D−01 0.253569D−01 0.239158D−01 0.223483D−01
4.20 0.266176D−01 0.254649D−01 0.241847D−01 0.227815D−01 0.212610D−01
4.30 0.254771D−01 0.243454D−01 0.230940D−01 0.217278D−01 0.202527D−01
4.40 0.244109D−01 0.233005D−01 0.220776D−01 0.207473D−01 0.193159D−01
4.50 0.234128D−01 0.223236D−01 0.211286D−01 0.198332D−01 0.184439D−01
4.60 0.224768D−01 0.214088D−01 0.202413D−01 0.189798D−01 0.176310D−01
4.70 0.215980D−01 0.205510D−01 0.194103D−01 0.181816D−01 0.168717D−01
4.80 0.207717D−01 0.197454D−01 0.186309D−01 0.174340D−01 0.161616D−01
4.90 0.199937D−01 0.189879D−01 0.178990D−01 0.167328D−01 0.154963D−01
5.00 0.192603D−01 0.182746D−01 0.172106D−01 0.160741D−01 0.148722D−01
t β
0.80 0.82 0.84 0.86 0.88
0.25 0.255413D−07
0.26 0.477184D−06
0.27 0.525457D−05
0.28 0.379878D−04 0.126195D−08
0.29 0.195935D−03 0.393014D−07
0.30 0.769172D−03 0.661087D−06
0.31 0.241777D−02 0.679523D−05
0.32 0.633452D−02 0.469762D−04
0.33 0.142827D−01 0.235440D−03 0.134598D−07
0.34 0.284347D−01 0.907667D−03 0.277183D−06
0.35 0.510312D−01 0.282134D−02 0.338285D−05
0.36 0.839632D−01 0.734202D−02 0.270323D−04
0.37 0.128408D+00 0.164880D−01 0.153019D−03
0.38 0.184615D+00 0.327469D−01 0.653185D−03 0.211304D−07
0.39 0.251870D+00 0.586822D−01 0.221037D−02 0.429655D−06
0.40 0.328623D+00 0.964459D−01 0.617305D−02 0.519765D−05
0.41 0.412712D+00 0.147350D+00 0.146982D−01 0.412273D−04
0.42 0.501631D+00 0.211615D+00 0.306364D−01 0.231645D−03
0.43 0.592782D+00 0.288329D+00 0.571221D−01 0.980769D−03 0.391931D−08
0.44 0.683687D+00 0.375601D+00 0.969762D−01 0.328848D−02 0.123064D−06
0.45 0.772133D+00 0.470836D+00 0.152115D+00 0.908959D−02 0.209758D−05
0.46 0.856271D+00 0.571057D+00 0.223145D+00 0.213979D−01 0.217958D−04
0.47 0.934646D+00 0.673206D+00 0.309230D+00 0.440573D−01 0.151403D−03
0.48 0.100620D+01 0.774397D+00 0.408235D+00 0.810860D−01 0.756786D−03
0.49 0.107025D+01 0.872089D+00 0.517052D+00 0.135811D+00 0.288761D−02
0.50 0.112644D+01 0.964190D+00 0.632014D+00 0.210091D+00 0.882059D−02
0.51 0.117466D+01 0.104910D+01 0.749308D+00 0.303876D+00 0.224174D−01
0.52 0.121504D+01 0.112569D+01 0.865316D+00 0.415177D+00 0.489103D−01
0.53 0.124788D+01 0.119327D+01 0.976859D+00 0.540408D+00 0.939817D−01
0.54 0.127358D+01 0.125155D+01 0.108134D+01 0.674938D+00 0.162409D+00
0.55 0.129264D+01 0.130052D+01 0.117680D+01 0.813706D+00 0.256789D+00
0.56 0.130560D+01 0.134044D+01 0.126191D+01 0.951773D+00 0.376807D+00
0.57 0.131301D+01 0.137174D+01 0.133590D+01 0.108473D+01 0.519216D+00
0.58 0.131544D+01 0.139498D+01 0.139849D+01 0.120895D+01 0.678437D+00
0.59 0.131345D+01 0.141080D+01 0.144980D+01 0.132171D+01 0.847512D+00
0.60 0.130757D+01 0.141988D+01 0.149023D+01 0.142118D+01 0.101910D+01
0.61 0.129829D+01 0.142291D+01 0.152043D+01 0.150636D+01 0.118632D+01
0.62 0.128607D+01 0.142059D+01 0.154116D+01 0.157692D+01 0.134336D+01
0.63 0.127135D+01 0.141356D+01 0.155330D+01 0.163310D+01 0.148574D+01
0.64 0.125450D+01 0.140245D+01 0.155772D+01 0.167555D+01 0.161045D+01
0.65 0.123589D+01 0.138785D+01 0.155534D+01 0.170523D+01 0.171579D+01
0.66 0.121583D+01 0.137028D+01 0.154702D+01 0.172324D+01 0.180120D+01
0.67 0.119460D+01 0.135023D+01 0.153357D+01 0.173081D+01 0.186705D+01
0.68 0.117244D+01 0.132813D+01 0.151577D+01 0.172918D+01 0.191436D+01
0.69 0.114958D+01 0.130438D+01 0.149432D+01 0.171957D+01 0.194460D+01
0.70 0.112621D+01 0.127931D+01 0.146985D+01 0.170315D+01 0.195951D+01
0.71 0.110249D+01 0.125324D+01 0.144294D+01 0.168100D+01 0.196094D+01
0.72 0.107858D+01 0.122643D+01 0.141408D+01 0.165410D+01 0.195076D+01
0.73 0.105459D+01 0.119910D+01 0.138373D+01 0.162336D+01 0.193074D+01
0.74 0.103064D+01 0.117146D+01 0.135226D+01 0.158956D+01 0.190258D+01
0.75 0.100681D+01 0.114369D+01 0.132002D+01 0.155340D+01 0.186780D+01
0.76 0.983178D+00 0.111592D+01 0.128730D+01 0.151549D+01 0.182776D+01
0.77 0.959818D+00 0.108828D+01 0.125433D+01 0.147635D+01 0.178366D+01
0.78 0.936781D+00 0.106087D+01 0.122133D+01 0.143642D+01 0.173656D+01
0.79 0.914109D+00 0.103379D+01 0.118848D+01 0.139608D+01 0.168733D+01
0.80 0.891840D+00 0.100711D+01 0.115592D+01 0.135565D+01 0.163673D+01
0.81 0.870002D+00 0.980871D+00 0.112377D+01 0.131539D+01 0.158540D+01
0.82 0.848618D+00 0.955136D+00 0.109214D+01 0.127552D+01 0.153386D+01
0.83 0.827706D+00 0.929938D+00 0.106109D+01 0.123622D+01 0.148254D+01
0.84 0.807276D+00 0.905305D+00 0.103069D+01 0.119762D+01 0.143177D+01
0.85 0.787338D+00 0.881259D+00 0.100100D+01 0.115984D+01 0.138185D+01
0.86 0.767897D+00 0.857814D+00 0.972040D+00 0.112296D+01 0.133297D+01
0.87 0.748953D+00 0.834979D+00 0.943843D+00 0.108704D+01 0.128531D+01
0.88 0.730507D+00 0.812759D+00 0.916425D+00 0.105213D+01 0.123899D+01
0.89 0.712555D+00 0.791156D+00 0.889796D+00 0.101827D+01 0.119408D+01
0.90 0.695094D+00 0.770166D+00 0.863958D+00 0.985464D+00 0.115066D+01
0.91 0.678116D+00 0.749785D+00 0.838910D+00 0.953725D+00 0.110874D+01
0.92 0.661615D+00 0.730004D+00 0.814646D+00 0.923050D+00 0.106835D+01
0.93 0.645582D+00 0.710816D+00 0.791156D+00 0.893431D+00 0.102948D+01
0.94 0.630007D+00 0.692210D+00 0.768428D+00 0.864854D+00 0.992123D+00
0.95 0.614883D+00 0.674173D+00 0.746446D+00 0.837299D+00 0.956246D+00
0.96 0.600197D+00 0.656693D+00 0.725195D+00 0.810745D+00 0.921820D+00
0.97 0.585939D+00 0.639756D+00 0.704657D+00 0.785167D+00 0.888808D+00
0.98 0.572099D+00 0.623348D+00 0.684812D+00 0.760537D+00 0.857168D+00
0.99 0.558665D+00 0.607456D+00 0.665641D+00 0.736827D+00 0.826855D+00
1.00 0.545627D+00 0.592064D+00 0.647125D+00 0.714008D+00 0.797822D+00
1.05 0.485978D+00 0.522098D+00 0.563645D+00 0.612226D+00 0.670177D+00
1.10 0.434586D+00 0.462487D400 0.493521D+00 0.528288D+00 0.567472D+00
1.15 0.390223D+00 0.411597D+00 0.434485D+00 0.458877D+00 0.484524D+00
1.20 0.351812D+00 0.368008D+00 0.384593D+00 0.401205D+00 0.417110D+00
1.25 0.318437D+00 0.330522D+00 0.342228D+00 0.353007D+00 0.361904D+00
1.30 0.289325D+00 0.298143D+00 0.306070D+00 0.312473D+00 0.316332D+00
1.35 0.263830D+00 0.270049D+00 0.275044D+00 0.278165D+00 0.278410D+00
1.40 0.241413D+00 0.245560D+00 0.248281D+00 0.248941D+00 0.246604D+00
1.45 0.221623D+00 0.224118D+00 0.225074D+00 0.223894D+00 0.219726D+00
1.50 0.204086D+00 0.205261D+00 0.204848D+00 0.202298D+00 0.196849D+00
1.60 0.174558D+00 0.173841D+00 0.171553D+00 0.167250D+00 0.160341D+00
1.70 0.150864D+00 0.148949D+00 0.145564D+00 0.140356D+00 0.132883D+00
1.80 0.131604D+00 0.128942D+00 0.124942D+00 0.119331D+00 0.111778D+00
1.90 0.115758D+00 0.112648D+00 0.108338D+00 0.102618D+00 0.952449D−01
2.00 0.102582D+00 0.992200D−01 0.947904D−01 0.891342D−01 0.820738D−01
2.10 0.915160D−01 0.880336D−01 0.836055D−01 0.781120D−01 0.714248D−01
2.20 0.821391D−01 0.786235D−01 0.742718D−01 0.689949D−01 0.627009D−01
2.30 0.741283D−01 0.706373D−01 0.664074D−01 0.613730D−01 0.554699D−01
2.40 0.672332D−01 0.638046D−01 0.597227D−01 0.549400D−01 0.494130D−01
2.50 0.612577D−01 0.579156D−01 0.539953D−01 0.494633D−01 0.442915D−01
2.60 0.560464D−01 0.528057D−01 0.490523D−01 0.447638D−01 0.399238D−01
2.70 0.514752D−01 0.483441D−01 0.447578D−01 0.407023D−01 0.361699D−01
2.80 0.474440D−01 0.444264D−01 0.410039D−01 0.371691D−01 0.329207D−01
2.90 0.438712D−01 0.409680D−01 0.377040D−01 0.340768D−01 0.300901D−01
3.00 0.406902D−01 0.379002D−01 0.347881D−01 0.313555D−01 0.276096D−01
3.10 0.378459D−01 0.351665D−01 0.321990D−01 0.289483D−01 0.254239D−01
3.20 0.352924D−01 0.327201D−01 0.298900D−01 0.268088D−01 0.234883D−01
3.30 0.329915D−01 0.305224D−01 0.278220D−01 0.248990D−01 0.217661D−01
3.40 0.309109D−01 0.285407D−01 0.259628D−01 0.231870D−01 0.202273D−01
3.50 0.290233D−01 0.267477D−01 0.242852D−01 0.216467D−01 0.188466D−01
3.60 0.273057D−01 0.251201D−01 0.227663D−01 0.202558D−01 0.176032D−01
3.70 0.257382D−01 0.236383D−01 0.213868D−01 0.189956D−01 0.164795D−01
3.80 0.243037D−01 0.222852D−01 0.201301D−01 0.178503D−01 0.154607D−01
3.90 0.229876D−01 0.210465D−01 0.189820D−01 0.168063D−01 0.145340D−01
4.00 0.217771D−01 0.199095D−01 0.179303D−01 0.158520D−01 0.136888D−01
4.10 0.206613D−01 0.188633D−01 0.169647D−01 0.149774D−01 0.129157D−01
4.20 0.196305D−01 0.178987D−01 0.160758D−01 0.141739D−01 0.122067D−01
4.30 0.186762D−01 0.170071D−01 0.152558D−01 0.134340D−01 0.115550D−01
4.40 0.177910D−01 0.161816D−01 0.144977D−01 0.127510D−01 0.109545D−01
4.50 0.169684D−01 0.154156D−01 0.137954D−01 0.121194D−01 0.104000D−01
4.60 0.162026D−01 0.147035D−01 0.131436D−01 0.115341D−01 0.988695D−02
4.70 0.154884D−01 0.140404D−01 0.125376D−01 0.109906D−01 0.941126D−02
4.80 0.148213D−01 0.134219D−01 0.119730D−01 0.104851D−01 0.896941D−02
4.90 0.141972D−01 0.128441D−01 0.114463D−01 0.100140D−01 0.855824D−02
5.00 0.136125D−01 0.123034D−01 0.109540D−01 0.957441D−02 0.817499D−02
t β
0.90 0.92 0.94 0.96 0.98
0.49 0.175529D−08
0.50 0.820397D−07
0.51 0.187578D−05
0.52 0.241437D−04
0.53 0.195463D−03
0.54 0.108692D−02
0.55 0.445324D−02
0.56 0.142197D−01 0.139846D−08
0.57 0.370207D−01 0.105583D−06
0.58 0.814990D−01 0.332932D−05
0.59 0.156250D+00 0.526634D−04
0.60 0.267214D+00 0.481944D−03
0.61 0.415676D+00 0.285267D−02
0.62 0.597664D+00 0.119205D−01
0.63 0.804762D+00 0.376784D−01
0.64 0.102585D+01 0.951387D−01
0.65 0.124906D+01 0.200409D+00 0.161725D−06
0.66 0.146347D+01 0.364567D+00 0.879749D−05
0.67 0.166018D+01 0.588766D+00 0.190728D−03
0.68 0.183281D+01 0.863064D+00 0.204369D−02
0.69 0.197757D+01 0.116905D+01 0.127375D−01
0.70 0.209293D+01 0.148449D+01 0.522574D−01
0.71 0.217915D+01 0.178792D+01 0.155091D+00
0.72 0.223785D+01 0.206190D+01 0.357914D+00
0.73 0.227150D+01 0.229453D+01 0.678961D+00
0.74 0.228302D+01 0.247956D+01 0.110508D+01 0.644329D−07
0.75 0.227556D+01 0.261551D+01 0.159526D+01 0.159988D−04
0.76 0.225220D+01 0.270449D+01 0.209580D+01 0.793681D−03
0.77 0.221587D+01 0.275095D+01 0.255668D+01 0.125896D−01
0.78 0.216924D+01 0.276060D+01 0.294194D+01 0.887758D−01
0.79 0.211465D+01 0.273958D+01 0.323279D+01 0.350948D+00
0.80 0.205417D+01 0.269392D+01 0.342548D+01 0.916457D+00
0.81 0.198953D+01 0.262917D+01 0.352689D+01 0.177633D+01
0.82 0.192220D+01 0.255023D+01 0.354994D+01 0.277714D+01
0.83 0.185338D+01 0.246128D+01 0.350992D+01 0.371726D+01
0.84 0.178405D+01 0.236580D+01 0.342203D+01 0.444662D+01
0.85 0.171500D+01 0.226658D+01 0.329993D+01 0.490323D+01 0.569651D−07
0.86 0.164683D+01 0.216587D+01 0.315514D+01 0.509740D+01 0.471182D−03
0.87 0.158002D+01 0.206538D+01 0.299689D+01 0.507860D+01 0.642527D−01
0.88 0.151493D+01 0.196642D+01 0.283231D+01 0.490757D+01 0.902888D+00
0.89 0.145181D+01 0.186995D+01 0.266669D+01 0.464012D+01 0.358054D+01
0.90 0.139086D+01 0.177666D+01 0.250383D+01 0.432050D+01 0.700029D+01
0.91 0.133218D+01 0.168701D+01 0.234632D+01 0.398056D+01 0.921090D+01
0.92 0.127584D+01 0.160128D+01 0.219584D+01 0.364151D+01 0.971591D+01
0.93 0.122186D+01 0.151963D+01 0.205339D+01 0.331644D+01 0.907158D+01
0.94 0.117025D+01 0.144210D+01 0.191947D+01 0.301261D+01 0.793277D+01
0.95 0.112096D+01 0.136868D+01 0.179422D+01 0.273340D+01 0.671051D+01
0.96 0.107396D+01 0.129928D+01 0.167752D+01 0.247970D+01 0.559410D+01
0.97 0.102917D+01 0.123378D+01 0.156910D+01 0.225091D+01 0.464461D+01
0.98 0.986518D+00 0.117203D+01 0.146856D+01 0.204556D+01 0.386369D+01
0.99 0.945935D+00 0.111387D+01 0.137548D+01 0.186177D+01 0.323073D+01
1.00 0.907332D+00 0.105912D+01 0.128936D+01 00.169753D+01 0.271997D+01
1.05 0.741019D+00 0.830273D+00 0.946793D+00 0.110334D+01 0.128811D+01
1.10 0.611719D+00 0.661154D+00 0.713482D+00 0.755736D+00 0.716959D+00
1.15 0.510612D+00 0.534903D+00 0.551345D+00 0.542306D+00 0.447586D+00
1.20 0.430831D+00 0.439309D+00 0.435858D+00 0.404585D+00 0.302938D+00
1.25 0.367219D+00 0.3658091+00 0.351547D+00 0.311677D+00 0.217435D+00
1.30 0.315955D+00 0.308434D+00 0.288562D+00 0.246554D+00 0.163107D+00
1.35 0.274204D+00 0.262995D+00 0.240516D+00 0.199394D+00 0.126609D+00
1.40 0.239857D+00 0.226524D+00 0.203171D+00 0.164281D+00 0.100986D+00
1.45 0.211332D+00 0.196886D+00 0.173652D+00 0.137506D+00 0.823451D−01
1.50 0.187431D+00 0.172528D+00 0.149968D+00 0.116666D+00 0.683815D−01
1.60 0.150048D+00 0.135340D+00 0.114859D+00 0.868623D−01 0.492615D−01
1.70 0.122586D+00 0.108773D+00 0.905940D−01 0.670597D−01 0.371293D−01
1.80 0.101890D+00 0.892001D−01 0.731814D−01 0.532722D−01 0.289650D−01
1.90 0.859419D−01 0.743997D−01 0.602917D−01 0.433063D−01 0.232152D−01
2.00 0.734156D−01 0.629564D−01 0.504986D−01 0.358786D−01 0.190167D−01
2.10 0.634102D−01 0.539376D−01 0.428925D−01 0.301997D−01 0.158591D−01
2.20 0.552997D−01 0.467104D−01 0.368724D−01 0.257635D−01 0.134256D−01
2.30 0.486391D−01 0.408340D−01 0.320290D−01 0.222338D−01 0.115110D−01
2.40 0.431054D−01 0.359941D−01 0.280763D−01 0.193803D−01 0.997794D−02
2.50 0.384601D−01 0.319622D−01 0.248098D−01 0.170415D−01 0.873155D−02
2.60 0.345241D−01 0.285691D−01 0.220801D−01 0.151009D−01 0.770466D−02
2.70 0.311609D−01 0.256874D−01 0.197761D−01 0.134733D−01 0.684867D−02
2.80 0.282652D−01 0.232197D−01 0.178142D−01 0.120950D−01 0.612771D−02
2.90 0.257547D−01 0.210907D−01 0.161300D−01 0.109177D−01 0.551483D−02
3.00 0.235642D−01 0.192415D−01 0.146736D−01 0.990411D−02 0.498948D−02
3.10 0.216417D−01 0.176251D−01 0.134059D−01 0.902539D−02 0.453577D−02
3.20 0.199455D−01 0.162043D−01 0.122956D−01 0.825862D−02 0.414124D−02
3.30 0.184415D−01 0.149487D−01 0.113178D−01 0.758561D−02 0.379605D−02
3.40 0.171017D−01 0.138338D−01 0.104523D−01 0.699167D−02 0.349229D−02
3.50 0.159032D−01 0.128394D−01 0.968251D−02 0.646491D−02 0.322361D−02
3.60 0.148268D−01 0.119487D−01 0.899487D−02 0.599557D−02 0.298479D−02
3.70 0.138565D−01 0.111478D−01 0.837809D−02 0.557560D−02 0.277158D−02
3.80 0.129788D−01 0.104250D−01 0.782277D−02 0.519832D−02 0.258044D−02
3.90 0.121823D−01 0.977054D−02 0.732102D−02 0.485814D−02 0.240842D−02
4.00 0.114572D−01 0.917606D−02 0.686616D−02 0.455034D−02 0.225307D−02
4.20 0.101896D−01 0.813959D−02 0.607530D−02 0.401658D−02 0.198432D−02
4.40 0.912234D−02 0.727000D−02 0.541399D−02 0.357168D−02 0.176098D−02
4.60 0.821535D−02 0.653328D−02 0.485541D−02 0.319696D−02 0.157337D−02
4.80 0.743806D−02 0.590366D−02 0.437933D−02 0.287840D−02 0.141425D−02
5.00 0.676682D−02 0.536134D−02 0.397025D−02 0.260530D−02 0.127813D−02
t β
0.990 0.995 0.997 0.998 0.999
0.9250 0.304435D−01
0.9300 0.724474D+00
0.9350 0.413781D+01
0.9400 0.103257D+02
0.9450 0.159373D+02
0.9460 0.167331D+02
0.9470 0.173985D+02
0.9480 0.179335D+02
0.9490 0.183414D+02
0.9500 0.186282D+02
0.9510 0.188019D+02
0.9520 0.188719D+02
0.9530 0.188481D+02
0.9540 0.187412D+02
0.9550 0.185613D+02
0.9560 0.183187D+02
0.9570 0.180228D+02
0.9580 0.176827D+02 0.207732D−01
0.9590 0.173064D+02 0.109476D+00
0.9600 0.169014D+02 0.413897D+00
0.9610 0.164744D+02 0.119543D+01
0.9620 0.160311D+02 0.277546D+01
0.9630 0.155769D+02 0.539843D+01
0.9640 0.151160D+02 0.909628D+01
0.9650 0.146524D+02 0.136434D+02
0.966G 0.141892D+02 0.186215D+02
0.9670 0.137293D+02 0.235457D+02
0.9680 0.132750D+02 0.279852D+02
0.9690 0.128279D+02 0.316365D+02
0.9700 0.123898D+02 0.343449D+02
0.9710 0.119618D+02 0.360858D+02
0.9720 0.115447D+02 0.369291D+02
0.9730 0.111393D+02 0.370001D+02 0.224193D−01
0.9740 0.107461D+02 0.364475D+02 0.327681D+00
0.9750 0.103653D+02 0.354200D+02 0.209692D+01
0.9760 0.999722D+01 0.340534D+02 0.747066D+01
0.9770 0.964186D+01 0.324632D+02 0.175707D+02
0.9780 0.929917D+01 0.307431D+02 0.307895D+02
0.9790 0.896903D+01 0.289656D+02 0.438034D+02
0.9800 0.865126D+01 0.271853D+02 0.537808D+02
0.9805 0.849694D+01 0.263068D+02 0.571903D+02
0.9810 0.834562D+01 0.254412D+02 0.595157D+02
0.9815 0.819727D+01 0.245914D+02 0.608249D+02 0.160180D+00
0.9820 0.805183D+01 0.237599D+02 0.612297D+02 0.889475D+00
0.9825 0.790929D+01 0.229486D+02 0.608644D+02 0.326711D+01
0.9830 0.776958D+01 0.221588D+02 0.598693D+02 0.871315D+01
0.9835 0.763267D+01 0.213917D+02 0.583798D+02 0.181354D+02
0.9840 0.749853D+01 0.206479D+02 0.565191D+02 0.311551D+02
0.9845 0.736709D+01 0.199279D+02 0.543950D+02 0.461344D+02
0.9850 0.723832D+01 0.192318D+02 0.520981D+02 0.609020D+02
0.9855 0.711217D+01 0.185598D+02 0.497029D+02 0.735681D+02
0.9860 0.698860D+01 0.179116D+02 0.472686D+02 0.829855D+02
0.9865 0.686755D+01 0.172869D+02 0.448414D+02 0.887989D+02
0.9870 0.674899D+01 0.166854D+02 0.424560D+02 0.912470D+02
0.9875 0.663286D+01 0.161067D+02 0.401379D+02 0.909006D+02
0.9880 0.651912D+01 0.155500D+02 0.379049D+02 0.884433D+02
0.9885 0.640773D+01 0.150149D+02 0.357690D+02 0.845305D+02
0.9890 0.629863D+01 0.145007D+02 0.337370D+02 0.797185D+02
0.9895 0.619178D+01 0.140068D+02 0.318125D+02 0.744417D+02 0.128057D−02
0.9900 0.608714D+01 0.135325D+02 0.299960D+02 0.690183D+02 0.253893D+00
0.9901 0.606647D+01 0.134399D+02 0.296456D+02 0.679369D+02 0.542909D+00
0.9902 0.604589D+01 0.133480D+02 0.292994D+02 0.668599D+02 0.107406D+01
0.9903 0.602539D+01 0.132570D+02 0.289575D+02 0.657887D+02 0.198062D+01
0.9904 0.600498D+01 0.131666D+02 0.286198D+02 0.647244D+02 0.342758D+01
0.9905 0.598466D+01 0.130770D+02 0.282862D+02 0.636679D+02 0.560072D+01
0.9906 0.596442D+01 0.129881D+02 0.279569D+02 0.626202D+02 0.868902D+01
0.9907 0.594426D+01 0.128999D+02 0.276316D+02 0.615821D+02 0.128629D+02
0.9908 0.592419D+01 0.128125D+02 0.273105D+02 0.605543D+02 0.182520D+02
0.9909 0.590420D+01 0.127258D+02 0.269934D+02 0.595376D+02 0.249263D+02
0.9910 0.588429D+01 0.126397D+02 0.266804D+02 0.585324D+02 0.328840D+02
0.9911 0.586447D+01 0.125544D+02 0.263713D+02 0.575394D+02 0.420476D+02
0.9912 0.584473D+01 0.124698D+02 0.260663D+02 0.565589D+02 0.522680D+02
0.9913 0.582507D+01 0.123858D+02 0.257652D+02 0.555914D+02 0.633368D+02
0.9914 0.580549D+01 0.123026D+02 0.254679D+02 0.546371D+02 0.750017D+02
0.9915 0.578600D+01 0.122200D+02 0.251746D+02 0.536965D+02 0.869862D+02
0.9916 0.576658D+01 0.121381D+02 0.248850D+02 0.527696D+02 0.990079D+02
0.9917 0.574725D+01 0.120569D+02 0.245993D+02 0.518567D+02 0.110795D+03
0.9918 0.572799D+01 0.119763D+02 0.243173D+02 0.509580D+02 0.122101D+03
0.9919 0.570882D+01 0.118964D+02 0.240390D+02 0.500735D+02 0.132714D+03
0.9920 0.568973D+01 0.118171D+02 0.237644D+02 0.492033D+02 0.142460D+03
0.9921 0.567071D+01 0.117385D+02 0.234934D+02 0.483476D+02 0.151211D+03
0.9922 0.565178D+01 0.116605D+02 0.232260D+02 0.475062D+02 0.158879D+03
0.9923 0.563292D+01 0.115832D+02 0.229621D+02 0.466793D+02 0.165415D+03
0.9924 0.561415D+01 0.115065D+02 0.227018D+02 0.458667D+02 0.170805D+03
0.9925 0.559545D+01 0.114304D+02 0.224449D+02 0.450684D+02 0.175065D+03
0.9926 0.557682D+01 0.113549D+02 0.221914D+02 0.442844D+02 0.178235D+03
0.9927 0.555828D+01 0.112801D+02 0.219413D+02 0.435146D+02 0.180374D+03
0.9928 0.553981D+01 0.112058D+02 0.216946D+02 0.427588D+02 0.181552D+03
0.9929 0.552142D+01 0.111322D+02 0.214512D+02 0.420169D+02 0.181851D+03
0.9930 0.550311D+01 0.110592D+02 0.212110D+02 0.412888D+02 0.181356D+03
0.9931 0.548487D+01 0.109867D+02 0.209741D+02 0.405744D+02 0.180155D+03
0.9932 0.546671D+01 0.109149D+02 0.207403D+02 0.398734D+02 0.178335D+03
0.9933 0.544862D+01 0.108436D+02 0.205097D+02 0.391858D+02 0.175979D+03
0.9934 0.543061D+01 0.107729D+02 0.202821D+02 0.385114D+02 0.173166D+03
0.9935 0.541267D+01 0.107028D+02 0.200577D+02 0.378500D+02 0.169971D+03
0.9936 0.539481D+01 0.106333D+02 0.198362D+02 0.372013D+02 0.166463D+03
0.9937 0.537702D+01 0.105643D+02 0.196177D+02 0.365653D+02 0.162703D+03
0.9938 0.535930D+01 0.104959D+02 0.194022D+02 0.359417D+02 0.158747D+03
0.9939 0.534166D+01 0.104281D+02 0.191895D+02 0.353303D+02 0.154646D+03
0.9940 0.532409D+01 0.103608D+02 0.189798D+02 0.347310D+02 0.150444D+03
0.9942 0.528917D+01 0.102278D+02 0.185686D+02 0.335676D+02 0.141883D+03
0.9944 0.525454D+01 0.100970D+02 0.181685D+02 0.324499D+02 0.133313D+03
0.9946 0.522019D+01 0.996825D+01 0.177790D+02 0.313762D+02 0.124914D+03
0.9948 0.518613D+01 0.984161D+01 0.173999D+02 0.303450D+02 0.116809D+03
0.9950 0.515235D+01 0.971701D+01 0.170309D+02 0.293546D+02 0.109080D+03
0.9960 0.498757D+01 0.912334D+01 0.153274D+02 0.249607D+02 0.770202D+02
0.9970 0.482945D+01 0.857550D+01 0.138362D+02 0.213653D+02 0.550465D+02
0.9980 0.467770D+01 0.806958D+01 0.125283D+02 0.184123D+02 0.403125D+02
0.9990 0.453203D+01 0.760198D+01 0.113782D+02 0.159741D+02 0.303186D+02
1.0000 0.439217D+01 0.716940D+01 0.103644D+02 0.139490D+02 0.233847D+02
1.0010 0.425787D+01 0.676883D+01 0.946812D+01 0.122560D+02 0.184504D+02
1.0020 0.412887D+01 0.639755D+01 0.867348D+01 0.108317D+02 0.148511D+02
1.0030 0.400493D+01 0.605304D+01 0.796691D+01 0.962565D+01 0.121648D+02
1.0040 0.388584D+01 0.573305D+01 0.733683D+01 0.859805D+01 0.101177D+02
1.0050 0.377138D+01 0.543552D+01 0.677335D+01 0.771724D+01 0.852845D+01
1.0100 0.326150D+01 0.422493D+01 0.469747D+01 0.478245D+01 0.422422D+01
1.0200 0.248913D+01 0.271963D+01 0.258091D+01 0.229363D+01 0.162193D+01
1.0300 0.194690D+01 0.187297D+01 0.160578D+01 0.132261D+01 0.841468D+00
1.0400 0.155592D+01 0.135820D+01 0.108674D+01 0.853819D+00 0.511577D+00
1.0500 0.126694D+01 0.102519D+01 0.780848D+00 0.594368D+00 0.342832D+00
1.1000 0.557631D+00 0.365425D+00 0.247198D+00 0.175463D+00 0.935803D−01
1.2000 0.192128D+00 0.109266D+00 0.691426D−01 0.473530D−01 0.243272D−01
1.3000 0.949409D−01 0.513358D−01 0.317892D−01 0.215305D−01 0.109368D−01
1.4000 0.561814D−01 0.296348D−01 0.181638D−01 0.122385D−01 0.618439D−02
1.5000 0.370172D−01 0.192470D−01 0.117281D−01 0.787899D−02 0.396971D−02
1.6000 0.261897D−01 0.134914D−01 0.819031D−02 0.549201D−02 0.276190D−02
1.7000 0.194902D−01 0.997588D−02 0.604054D−02 0.404529D−02 0.203175D−02
1.8000 0.150625D−01 0.767343D−02 0.463770D−02 0.310293D−02 0.155700D−02
1.9000 0.119859D−01 0.608435D−02 0.367208D−02 0.245513D−02 0.123108D−02
2.0000 0.976264D−02 0.494191D−02 0.297928D−02 0.199084D−02 0.997720D−03
2.2000 0.683487D−02 0.344570D−02 0.207391D−02 0.138473D−02 0.693413D−03
2.4000 0.504998D−02 0.253861D−02 0.152623D−02 0.101848D−02 0.509728D−03
2.6000 0.388252D−02 0.194762D−02 0.116996D−02 0.780411D−03 0.390419D−03
2.8000 0.307752D−02 0.154129D−02 0.925282D−03 0.617008D−03 0.308575D−03
3.0000 0.249916D−02 0.125002D−02 0.750041D−03 0.500026D−03 0.250008D−03
3.2000 0.206974D−02 0.103414D−02 0.620249D−03 0.413413D−03 0.206660D−03
3.4000 0.174220D−02 0.869712D−03 0.521449D−03 0.347500D−03 0.173682D−03
3.6000 0.148669D−02 0.741601D−03 0.444507D−03 0.296181D−03 0.148011D−03
3.8000 0.128354D−02 0.639848D−03 0.383419D−03 0.255445D−03 0.127637D−03
4.0000 0.111937D−02 0.557690D−03 0.334112D−03 0.222570D−03 0.111199D−03
4.2000 0.984811D−03 0.490399D−03 0.293739D−03 0.195657D−03 0.977428D−04
4.4000 0.873143D−03 0.434595D−03 0.260267D−03 0.173346D−03 0.865894D−04
4.6000 0.779455D−03 0.387803D−03 0.232207D−03 0.154645D−03 0.772418D−04
4.8000 0.700083D−03 0.348183D−03 0.208453D−03 0.138815D−03 0.693300D−04
5.0000 0.632254D−03 0.314341D−03 0.188167D−03 0.125297D−03 0.625745D−04

Figure 1a.

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.

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.

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 esβ

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.

Figure 2a

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

Figure 2b.

Figure 2b

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

Figure 3.

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.

4. Acknowledgment

Menachem Dishon is grateful for a position as a guest scientist at the Applied and Computational Mathematics Division of the National Institute of Standards and Technology, while on sabbatical leave from the Ministry of Defense, Tel Aviv, Israel. All computations were performed on the VAX computing facilities of the Center for Computing and Applied Mathematics at NIST.

Biography

About the authors: Menachem Dishon, an applied mathematician, served as guest scientist in NIST’s Applied and Computational Mathematics Division. He has returned to his position as advisor on computerization at the Ministry of Defense, Hakirya, Tel Aviv, Israel. John T. Bendler is staff physicist at the Polymer Physics and Engineering Branch of GEC Corporate Research and Development. George H. Weiss is Chief of the Physical Sciences Laboratory, Division of Computer Research and Technology, at NIH.

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