Tables of the Inverse Laplace Transform of the Function e−sβ

The inverse transform, g(t)=L−1(e−sβ), 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).

The inverse transform, g(/) = Jf ~'(e~'^), 0<;3<1, is a stable law that arises in a number of different applications in chemical physics, polymer physics, solid-state pliysics, 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 fi between 0.25 and 0.999. In addition, since g{t), regarded as a function of /8, is unimodal with a peak occurring at t = tn^" we both tabulate and graph fmai and l/5('max) as a function of /3, as well as giving polynomial approximations to 1/

Introduction
It has been known for at least 150 years that mechanical relaxation in solids is non-exponential, the decay often being characterized by a fractional power-law or logarithmic function [1,2]. It is also now generally recognized that all glassy materials exhibit non-exponential relaxation behavior both above and below the glass transition temperature, Tg. This is especially clear from measurements obtained from mechanical [3][4][5][6], dielectric [7][8][9], and photon correlation spectroscopy [10,11]. It is also seen in measurements of volumetric [12], and thermal response [13,14], In recent years theorists have become interested in the possibility that complex disordered systems exhibit universal features in their relaxation and transport properties, possibly arising from self-similar arrangements of obstacles to motion. This has been particularly encouraged by the observation that nearly all glassy relaxation phenomena can be described by the Kohlrausch-Williams-Watts (KWW) function In many physical applications it is convenient to represent <^(/;T) in the form of a Laplace transform, which we write as

(f)(?;T)= rV(M) e-" du = r"e-" A,(v) dv (2)
Jo Jo where (3) Thus, the function hr(y) can be found as an inverse Laplace transform of the function <l)(t;r). The function ^^(v) has found appHcation 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 hr(y) [20][21][22][23][24][25], in addition to a representation of hr(y) in terms of a convergent series given by Pollard [26]. Without loss of generality we can set T=1 since A^Cv) can be represented in terms of the inverse transform ^^<^>=2^-J/ !-^, ds (4) where F is a line to the right of the origin and parallel to the imaginary axis. The convergent series given by Pollard is (5) In an earlier paper we have presented an accurate tabulation of the sine and cosine transforms of the function exp{-t^), needed for the analysis of measurements of dielectric properties taken as a function of frequency [27]. In the present paper we tabulate the inverse Laplace transform hi(y). These tables may be used directly for the analysis of experimental data, but are also intended for use as a check on more easily programmed approximations, such as those suggested by earlier investigators [28][29][30].

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 tianstoim g(s)=Jf{g(t)} can be expressed in the form of a Fourier series: R e-U(« +i -jr)J-cos( -^r) -/m{g(a-|-/^)}sin(^)]) (6) with an error, E{t)=g^(t)-g(t), given by The function g(a -\-i-jr) can be written in terms of the parameters

bk=kTT/T, rk=y/a^+bL dk=tan-\bk/a)
as -i sin^/-|sinO80,)) (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 (5)=exp(-s^), the choice of these parameters is quite straightforward as will be shown below. Equation (6) was used to evaluate the inverse transform of g{s) for values of /3 in the range 0.20</3<0.999 and values of t ranging from lO""^ (for selected values of /3) to 5, to an accuracy of at least nine significant digits. In these ranges of i8 and t the choice of parametric ranges a €(2.5,5) and 7'e (4,8) sufficed to produce the stated accuracy. The accuracy of the numerical inversion can be checked in detail for three cases in which the inverse transforms are known exactly. where Ai(A:) 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 g{s) for ^=1/3, 1/2, and 2/3 for different values of a and T 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 A:<22 as well as a Pade correction to Stirling's approximation at larger k [33]. Thus, we write r(l+;8A:) where '^ Pik) (10)

Tables, Graphs, and Numerical Approximations
The inverse transform of the function g{s) is tabulated in table 2 for the following values of /?: 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 /? at low values of /3 are required because of the considerable changes in the function in that neighborhood. Spacings in t vary with fi 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 f >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 ji in the interval (0.05,0.999). For example, if ^8=0.6 the sum of seven terms of the series gives g (10) to six places, and the sum of four terms gives g(lOO) to the same accuracy. Figures la-c contain graphs of g(f) as a function of t over the entire range of tabulated values of /3. Note that for ;8 = 1 g{t)=8(t -1), a Dirac delta function which is represented as a vertical line in figure Ic.
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 CM- Table 3 contains some values of /max and g(/n,ax) for the values of yS for which we performed our tabulations. It is interesting to observe that among the values of g(/max) there is a minimum value within the interval (0,1). Figure 2a shows graphs of /^ax and l/g(/max) as functions of )3 for values of ^8 between 0.15 and 1. The minimum of g(?max) occurs at /"ax=0.252-h and is equal to 0.888-1-. These values correspond to )8=0.567-t-. Figure 2b contains a plot of l/g(/max) as a function of /max-Finally, we have derived polynomial leastsquare approximations to l/g(/max) as a function of /3. 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 l/g(/max) probably requires fitting some function other than a polynomial.

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.