Gaussian Quadrature Involving Einstein and Fermi Functions With an Application to Summation of Series*

Transcription

1 MATHEMATICS OF COMPUTATION VOLUME 44, NUMBER 169 JANUARY 1985, PAGES Gaussian Quadrature Involving Einstein and Fermi Functions With an Application to Summation of Series* By Walter Gautschi and Gradimir V. Milovanovic Abstract. Polynomials trk() = trk(; d\), k = 0,1,2,..., are constructed which are orthogonal with respect to the weight distributions d\(t) = (t/(e' - l))r dt and d\(t) = (I/O' + V)Y dt, r = 1,2, on (0, oo). Moment-related methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients ak, ßk in the recursion formula " t+i(0 = (' - «*K(0 - /V*-i(0. * - 0,1,2,..., ir0(f) = 1, «_!(<) - 0. The discretization is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively. Numerical values of ak, ßk, as well as associated error constants, are provided for 0 < k ^ 39. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of n-point formulae, n = 5(5)40, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series. 1. Introduction. We are interested in Gaussian quadrature on [0, oo] relative to the weight functions er(t) = (t/(e' - l))r and <pr(f) = (l/(e' + l))r. These functions arise, for example, in solid state physics and are referred to, when r = 1, as Einstein and Fermi functions, respectively. Integrals with respect to the measure dx(t) = er(t) dt, r = 1 and 2, are widely used in phonon statistics and lattice specific heats [7, 10], [1, 2.4], and occur also in the study of radiative recombination processes [9, 9.2]. Specifically, the average energy of a quantum harmonic oscillator of frequency «at temperature T (representing thermal vibrations of crystal lattice atoms) is given by h< (1.1) u = exp(hu/kt) - 1 ' where h = h/2m (h is the Planck constant) and k is the Boltzmann constant. Therefore, U = ktex(hu/kt), where ex is Einstein's function. Letting g(u) denote the phonon density of states function, and integrating (1.1) over the entire frequency range, the total energy of thermal vibration of the crystal lattice becomes Received April 11, 1983; revised November 14, 1983 and April 3, Mathematics Subject Classification. Primary 33A65, 65D32; Secondary 65A05, 65B10. * The work of the first author was sponsored in part by the National Science Foundation under grant MCS A American Mathematical Society /85 $ $.25 per page

2 178 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Furthermore, differentiating U with respect to temperature T, yields the crystal lattice heat capacity at constant volume, - = f (hu/kt)2 ;kg(o>) exp(hu/kt) 0 (exp(r2w//cr) - 1) du, that is, (1.3) C = k2t / <» / CO / e2(t)f(t) dt where f(t) = e'g((kt/h)t). Similarly, integrals of the type /0 q>x(t)f(t) dt (in fact, more generally /o <Pi(* ~ to)f(t) dt, for some parameter t0) are encountered in the dynamics of electrons in metals, where they express the specific heat of the electron gas [8, 9.17]. For an application to heavy doped semiconductors, see also [10]. It will be shown in Section 4 that integration with respect to dx(t) = ex(t) dt and dx(t) = <px(t) dt is also useful in summing slowly convergent series whose general term is expressible in terms of a Laplace transform or its derivative. Gaussian quadrature relative to the weight functions er and <pr will be particularly attractive in the case r = 1, since both weight functions then have poles with nonzero residues, the former at 2mri, n = ±1, ±2,..., the latter at (2n + l)wi, n = 0, +1, ±2,_Classical integration, even based on Gauss-Laguerre quadrature, will often be too slow, but will be used to generate the desired quadrature rules, at least in the case of the weight function er; see (2.3) below. The problem, basically, amounts to generating the coefficients ak, ßk in the recursion formula «*+i(0 - ('- «*K(0 - rv*-i(')> k = 0,1,2,..., n_x(t) = 0, *0(0 = 1 for the (monic) orthogonal polynomials trk(-) = irk(-; dx), k = 0,1,2, (/?0 is arbitrary, but will be defined as ß0 = /0 dx(t).) Once the first n of these coefficients, ak, ßk, k = 0,1,...,n - 1, and ßn, are known, the w-point Gaussian quadrature formula (1.4) / CO / f(t)dx(t)= V(t ) + *m(/)> ^0,1-1 Rjf) = yj(2m)(r), o<t<^, including the error constant ym, can easily be obtained for any m with 1 < m < n. The nodes tm = r m), indeed, are the eigenvalues of the (symmetric tridiagonal) Jacobi matrix /_.= Wi «1 {K 0 ypm-l am-\

3 GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 179 while the weights X^ = X^"1' are given by X^ = ß0v2x in terms of the first components v^ of the corresponding normalized eigenvectors; see, e.g., [3, 5.1]. Moreover, /Vi ft, (2m)! 1,2,...,«. A number of methods are available, and have been analyzed in [4], for computing the recursion coefficients ak, ßk. They are either based on the moments (or modified moments) of the weight distribution dx(t), or else use a suitable discretization of the inner product / OO (u,v)=f u(t)v(t) dx(t), u,v Jn P. where u, v are polynomials. In the next section we discuss the relative merits of these methods for constructing the particular orthogonal polynomials at hand. 2. Generation of the Recursion Coefficients. We consider first the weight function er. The classical approach (for example, the Chebyshev algorithm; see [4, 2.3]) departs from the moments of the weight function, which, in the case at hand, can be expressed in terms of the Riemann zeta function f(z), / OO M* = / Jo t\(t) dt (2.1) (k + 1)! (* + 2), r=l, (* + 2)![?(* + 2)- (* + 3)], r = 2, Jfc = 0,1,2,. It is well-known, however, that the moments pk define the recursion coefficients <xk, ßk and the associated Gaussian quadrature formulae poorly in a numerical sense. The map Gn from the first 2«moments pk,k = 0,\,2,...,2n - 1, to the weights X'J1' and nodes T (n) of the «-point Gaussian formula, in fact becomes progressively more ill-conditioned as n increases. This is illustrated in Table 2.1, where in the second and third column the condition number of this map (in the sense of [4, Eq. (3.1 l)f]) is shown as a function of «for r = 1 and r = 2, respectively. (Numbers in parentheses indicate decimal exponents.) Table 2.1 The numerical condition of various maps relating to the construction of orthogonal polynomials irk( ; erdt), r = 1, cond Gn r= (2) 1.288(5) 9.092(7) 6.334(10) 4.409(13) 3.076(16) cond Gn r = (1) 7.171(4) 5.820(7) 4.471(10) 3.344(13) 2.465(16) cond G r= (1) 2.934(3) 9.795(5) 4.039(8) 1.878(11) 9.441(13) cond Hn r= (0) 2.163(1) 3.299(1) 4.256(1) 6.761(1) 8.531(1) cond Hn r = (0) 2.161(1) 3.356(1) 4.244(1) 6.783(1) 8.556(1)

4 180 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Sometimes the condition can be improved by employing modified moments vk = j pk(t)er(t) dt, where {pk} is a system of (monic) polynomials suitably chosen. If the support interval is infinite, however, as in the case at hand, the improvement is often not sufficient to make this a viable alternative. The fourth column of Table 2.1 indeed confirms this. Here, r = 1, and ex(t) ~ te'' as t -» oo, so that the monic Laguerre polynomials with parameter a 1, pk(t) = (-1)**!L >(/), k- 0,1,2,..., appear to be the most natural choices for the polynomials pk. The corresponding modified moments can be shown to be expressible in terms of the (forward) differences, with unit step, of the Riemann zeta function at z = 2, / OO (2.2) vk= "o pk(t)ex(t)dt=(k + \)\Akï(z)\^2, k = 0,1,2,... The map Gn in Table 2.1 is the map from the first 2«normalized modified moments *k = vk\\opk\t)ie'tdt]'x/2, k «0,1,...,2«- 1, to the weights X<;> and nodes t,(b) of the «-point Gaussian quadrature formula (1.4) (with dx(t) = ex(t) dt). The entries in the fourth column exhibit the estimate derived in [4, Eq. (3.15)] for the condition of G. It can be seen that cond G, while somewhat smaller than cond Gn, is still growing unacceptably fast. The modified Chebyshev algorithm (cf. [4, 2.4]), run in single precision on the CDC 6500 (machine precision: ca. 15 significant decimal digits), with the modified moments (2.2) computed in double precision, indeed loses accuracy very much in accordance with the growth of cond Gn; the number of correct decimal digits observed in ak, ßk, k = 3(3)18, is 14, 11, 10, 6,2,0, respectively, not a single digit being correct for k = 18, not even the sign of ßxs. An additional complication arises in connection with the high-order differences in (2.2), which are subject to considerable cancellation errors. For k = 40, for example, one must expect a loss of approx. 12 decimal digits. In contrast, methods based on discretization appear to be incomparably more effective, particularly if one uses the natural discretization based on the Gauss- Laguerre quadrature rule, (2.3) " fc-1 \l-e-rt/rj Here, rk are the zeros of the Laguerre polynomial of degree N, and XLk the associated Christoffel numbers. The discretized Stieltjes procedure (cf. [4, 2.2]), based on (2.3), then converges fairly rapidly as N -» oo. For example, to obtain the first 12 recursion coefficients («= 12) to 12 correct decimals [25 correct decimals] requires N = 49 [N = 127], when r = 1, and N = 41 [N = 85], when r = 2. The first 40 coefficients («= 40) can be obtained accurately to 25 decimal digits with N = 281 for r = 1 and N = 201 for r = 2. Also the numerical stability of the procedure,

5 GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 181 which is roughly determined (cf. [4, 3.4]) by the condition of the map Hn: (Xx,...,Xn,Tx,...,Tn) -» (a0,...,an_x,ß0,...,ß _x), is rather good. This can be seen from the last two columns of Table 2.1, which display the condition number of Hn in the sense of [4, Eq. (3.8)f]. One might expect that a discretization analogous to the one in (2.3) works well also for the weight function <pr. This, however, is not the case, at least not if r = 1. Convergence turns out to be slow on account of the poles at ± iir, which are twice as close to the real axis as in the case of ex. For example, to obtain 20 correct decimal digits (in the case r 1 and for «= 40), requires N = 361. We used instead a discretization procedure based on the composite Fejér quadrature rule, decomposing the interval of integration into four subintervals, [0, oo] = [0,10] U [10,100] U [100,500] U [500, oo]. Using A* = 201 for r = 1, and N = 281 for r = 2, on each subinterval, then yields the first «= 40 recursion coefficients accurately to 25 decimal digits. The same procedure (with n = 50 and A* = 251) was also used to check the computation based on (2.3) for er, r = 1,2. The maximum discrepancy observed was 1 unit in the 25th digit. 3. Numerical Results. Using the procedures outlined in the last two paragraphs of Section 2, we obtained the results shown in Tables 1-4 of the Appendix. The coefficients ft0 = f(2) = ir2/6 in Table 1 and ft0 = 2[f(2) - f(3)] in Table 2 (cf. Eq. (2.1)), agree to all 25 decimal digits with the values obtained from the 41S-table of the Riemann zeta function in [6]. Similarly, one checks the values ft0 = In 2 in Table 3 and ft, = In 2 - \ in Table 4. Corresponding «-point Gaussian quadrature formulae, «= 5(5)40, for r = 1 and r = 2, can be found to 25 significant decimal digits in Tables 5-8 in the supplements section at the end of this issue. We illustrate these formulae by computing the integrals /i-/"«-'t7*-ï(2)-l, Jo e 1 h = jt e'\^h\\dt = 2^2>2) - 2f(3'2)]> r00. dt I3= I e'-= 1 -ln2, 3 h e'+l f dt 3 h= e-> -" 2 = -21n2, Jo (e'+l)2 2 where f(z, a) is the generalized zeta function. Table 3.1 shows the «-point approximations /,(«) to I, i = 1,2, together with the relative errors r (n), for n = 5(5)25. Similarly, /,(«) and r,(«), / = 3,4, are shown in Table 3.2. In each entry the first digit in error is underlined. The error term in (1.4) yields the bounds 7 //,, / = 1,2,3,4, for the relative error, which can be computed with the help of Tables 1-4 of the Appendix and the limit values in Tables 3.1, 3.2; they are summarized in Table 3.3 for «= 5(5)25. The bounds for I2,14 are seen to be considerably sharper than those for IX,I3.

6 182 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Table 3.1 Gaussian approximations of the integrals Ix and I2 and relative errors. /,(«) r,(n) I2(n) r2(n) (-4) l.l(-8) (-13) (-18) (-22) (-6) (-12) (-19) l.l(-25) Table 3.2 Gaussian approximations of the integrals I3 and I4 and relative errors. I3(n) r3(«) 74(«) r4(n) l-2(-4) (-9) (-14) (-18) (-23) (-7) (-13) (-20) (-26) Table 3.3 Relative error bounds for I (n),i = 1,2,3,4. 7n/h(") Y A(«) y /h(n) y /h(") 4.8(-2) 8.3(-5) 1.2(-2) 1.3(-5) l.l(-4) 3.0(-10) 1.6(-5) 1.9(-11) 1.9(-7) 6.7(-16) 2.0(-8) 2.3(-17) 2.7(-10) 1.2(-21) 2.3(-ll) 2.5(-23) 3.6(-13) 1.8(-27) 2.5(-14) 2.7(-29) 4. Summation of Series. As an application of Gaussian integration with weight functions e, and <px, we consider the summation of series whose general term is expressible in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. Suppose, for example, that ak = -F'(k), where (4.1) Then JfOO e-p'f(t)dt, n Re/7>1. that is, (4.2a) 00 r o r t Lak=L te-k'f(t)dt= -^ f(t)dt, :=i k-ij J e'-l k- ÎF'(k)=rf(t)ex(t)dt.,. 'n fc-1 'o Similarly, for "alternating" series, one obtains (4.2b) -t(-l)k-1f'(k)=ff(t)t<px(t) dt k = l

7 and (4.2c) GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 183 l(-l)k-1f(k)=rf(t)<px(t)dt. k-i Jo If the series on the left are slowly convergent and the respective function / on the right is smooth, then low-order Gaussian quadrature applied to the integrals on the right provides one (among many other) possible summation procedure. If / exhibits singularities, then the weight function may have to be modified if the effectiveness of the procedure is to be preserved (cf. Example 4.4). In the following examples, "(a)", "(b)", "(c)" refers to a, b, c of Eq. (4.2). Example 4.1. (a)!**_,(*: + l)"2 = 7r2/6-1. Here, -F'(p) = (p + l)-2, and F(p) = (P + I)-1) the integration constant being zero on account of F(p) -» 0 as p -» oo. Thus, f(t) = e'1, and the second column of Table 3.1 shows the rapid convergence of Gaussian quadrature in this case. For example, 15-point quadrature already yields 12 correct decimal digits. In contrast, terms of the series would give only 3-digit accuracy. (b) L"_1(-l)*~1(/c + l)"2 = 1 - tt2/12. Applying Gaussian quadrature to the integral in (4.2b) yields the results shown in Table 4.1. To guarantee, on the basis of Leibniz' convergence criterion, the same accuracy as the one achieved for n = 15 would require the summation of approximately terms. (c) Zf-x(-l)k~l(k + l)"1 = 1 - ln2. The convergence behavior of Gaussian quadrature applied to the integral in (4.2c) is evident from the second column of Table 3.2. Table 4.1 Gaussian approximations K(n) of the integral i"00 \-i K=j e~'t(e' + 1) dt and relative errors. n K(n) r(n) L2(-3) (-8) (-12) (-17) (-21) (-25) Example 4.2. F(p) = p-lcxp(-l/p), -F'(p) = (p - l)p~3 exp(-l/p). The original function is here /(f) = J0(2<JJ). This being an entire function, we expect Gaussian quadrature in (4.2) to converge rapidly. This is confirmed in Table 4.2, which shows the relative errors for the «-point formula, «= 2(2)12. The exact sums (to 24 significant digits), as determined by Gaussian quadrature, are, respectively, OO (k - l)k-3exp(-l/k) = , k = l oo (-l)k~\k - l)fc-3exp(-la) = , fc-i 00 E (-l)/c_1/i"1exp(-l/a:)= *: =!

8 184 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC The first terms of the series yield, respectively, 3,7 and 4 correct decimal digits. The Bessel function J0 was evaluated by means of the rational approximations in [5] indexed 5852, 6553 and Table 4.2 Relative errors in Gaussian approximation of the integrals in (4.2, a-c), wheref(p) _n_(a)_(b)_(c) = p~lexp(-\/p) (-2) 8.95(-l) 1.77(-2) (-6) 2.39(-4) 9.65(-7) (-ll) 3.71(-9) 6.32(-12) (-17) 1.13(-14) 1.05(-17) (-23) 1.08(-20) 1.27(-23) (-23) Example 4.3. F(p) = (1 + p2y1/2, -F'(p) = p(l + p2)~3/2. Here, f(t) = J0(t) again an entire function. Convergence of Gaussian quadrature in (4.2), while still satisfactory, is not quite as fast as in Example 4.2, presumably since the argument of JQ is now essentially squared. The relative errors are shown in Table 4.3. For reference, we list the exact values of the respective sums: 00 A:(l + k2)~3/2 = , *=i (-1) k(l + k2) ' = , k = l oo (-1) (1 + k2)\ ' = k = l The first terms yield 3,8 and 4 correct decimal digits, respectively. Table 4.3 Relative errors in Gaussian approximation of the integrals in (4.2, a-c), where F(p) _n_(a)_(b)_(c) = (l + p2)~l/ (-3) 8.15(-3) 1.99(-4) (-7) 4.75(-6) 2.11(-7) (-10) 2.99(-9) 9.01(-12) (-13) 7.61(-13) 2.26(-14) (-17) 5.64(-16) 2.52(-19) (-20) 8.25(-20) 2.33(-21) (-24) 8.01(-23) 3.88(-25)

9 GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 185 Example 4.4. F(p) = 2p'\p + 1)"1/2, -F'(p) = (3p + 2)p-2(p + l)"3/2. Here we have an example in which the original function, f(t) = 2 erf \ft, is no longer smooth, having a square root singularity at t = 0. Table 4.4 Relative errors in Gaussian approximation of the integrals in (4.2, a-c), where F(p) = 2p~x(p + 1)~1/2, using weight functions t1/2ex(t) andtl/2<px(t). n (a)_(b)_(c) (-5) 4.69(-5) 9.20(-6) (-10) 1.54(-9) 1.58(-10) (-14) 3.88(-14) 2.70(-15) (-19) 8.78(-19) 4.61(-20) (-24) 1.85(-23) 4.91(-25) This severely retards convergence of Gaussian quadrature in (4.2). When «= 40, for example, the relative errors are still 4.29(-4), 7.19(-6) and 4.80(-4), respectively. Better results can be obtained by integrating with respect to the measures t1/2ex(t) dt and ti/2(px(t) dt. The recursion coefficients for the corresponding orthogonal polynomials have been computed by a discretized Stieltjes procedure [4, 2.2] involving two different discretizations, one on the interval [0,10], where a Gauss-Jacobi quadrature with parameters a = 0, ft = \ (to account for the r1/2-singularity at t = 0) was used, and one on the interval [10, oo], where we used Gauss-Laguerre quadrature as before. With the special Gaussian quadrature rules thus obtained, convergence of the summation process is very satisfactory; see Table 4.4. The exact values of the sums are: oo (3A: + 2)k~2(k + l)~3/2 = , k = l oo (-l)*_1(3jfe + 2)k'2(k + \)'3/2 = , /c=l oo (-1)*"12*-1(A: + 1)"1/2 = fc-i The error function was evaluated by a double precison version of the incomplete gamma function routine in [2], observing that erf ]ft - g*(\, t)\n the notation of [2]. Acknowledgment. The authors are indebted to Professors R. Colella and N. D. Stqjadinovic for discussions of solid state physics applications and for the references [1], [7]-[10].

10 186 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Appendix AI: Recursion coefficients ak, ßk and error constants yk for orthogonal polynomials irk(-\ erdt), r = 1,2. Table 1 Recursion coefficients and error constants in the case r = 1 alpha(k) beta(k) gamma(k) d+0 0, d d d d d d d+01, d d d d d d+01, d+01, d+01, d+01, d+01, d+01, d+01, d+01, d+01, d+01, d+01, d+01, d d+01, d+01, d+01, lld+01, d+01, d+01, d d+01, d+01, d+01, d d+01, d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d-ll 2609d-ll 3531d d d d d d d d d d d d d d d d d-21

11 GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 187 Table 2 Recursion coefficients and error constants in the case r alpha (k) beta(k) gamma (k) d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d+01 1, d d+01 1, d d+01 2, lld d+01 2, d d+01 2, d d+01 2, d d+01 2, d d+01 2, d d+01 3, d d d+01 3, d d d d d d d d d d d d d d d d d d d d-ll d d d d d d d d d d d d d d d d d d d d d d d d d d d d-44

12 188 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Appendix A2: Recursion coefficients ak, ßk and error constants yk for orthogonal polynomials Trk(-;<prdt),r = 1,2. Table 3 Recursion coefficients and error constants in the case r = 1 alpha(k) beta(k) gamma(k) d+0O O97d d d d d d d d d d d O3O90O O524d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ld d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d-ll d d d d d d d d d d d d d d d d d d d d-23

13 GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 189 Table 4 Recursion coefficients and error constants in the case r alpha(k) beta(k) gamma(k) d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d O d d d d d d d d d d d d d d d d d d d d d d d d d d d d-ll d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d-47

14 190 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC Department of Computer Sciences Purdue University West Lafayette, Indiana Faculty of Electronic Engineering Department of Mathematics University of Ni NiS, Yugoslavia 1. J. S. Blakemore, Solid State Physics, 2nd ed., Saunders, Philadelphia, Pa., W. Gautschi, "Algorithm 542 Incomplete gamma functions," ACM Trans. Math. Software, v. 5, 1979, pp W. Gautschi, "A survey of Gauss-Christoffel quadrature formulae," in E. B. Christoffel The Influence of his Work in Mathematics and the Physical Sciences (P. L. Butzer and F. Fehér, eds.), Birkhäuser Verlag, Basel, 1981, pp W. Gautschi, "On generating orthogonal polynomials," SI AM J. Sei. Statist. Comput., v. 3, 1982, pp J. F. Hart et al., Computer Approximations, Wiley, New York-London-Sydney, A. McLellan IV, "Tables of the Riemann zeta function and related functions," Math. Comp., v. 22,1968, Review 69, pp S. S. Mitra & N. E. Massa, "Lattice vibrations in semiconductors," Chapter 3 in: Band Theory and Transport Properties (W. Paul, ed.), pp Handbook on Semiconductors (T. S. Moss, ed.), Vol. 1. North-Holland, Amsterdam, F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, R. A. Smith, Semiconductors, 2nd ed., Cambridge Univ. Press, Cambridge, R. J. Van Overstraeten, H. J. DeMan & R. P. Mertens, "Transport equations in heavy doped silicon," IEEE Trans. Electron Dev., v. ED-20, 1973, pp

15 mathematics of computation volume 44, number 169 JANUARY 1985, PAGES Sl-Sll Supplement to Gaussian Quadrature Involving Einstein and Fermi Functions with an Application to Summation of Series By Walter Gautschi and Gradimir V. Milovanovic TABLE 5. Nodes and weights for n-point Gaussian quadrature with respect to the weight function e,(t)=t/(e -1) on (0,»), n=5(s)40. zero(n) weight(n) O6d d+û d d d d d d d d-05 zerotn) weight(n) d d d d O d d d d d d d d d d d O O99d d d d d-12 zerotn) weight(n) d d d+O d d d d d d d d d d d d d d d d d d d d d d d d d d d-19 SI 1985 American Mathematical Society /85 $ $.25 per page

16 SUPPLEMENT CO O h r\i <f i p~ o f co co o o o CN CN CN co co co -cr I I I I l i i I I I I 1 i i i i t TJ TJ TJ *D TJ TJ TJ TJ TJ TJ TJ TJ r TJ TJ TJ TJ TJ TJ TJ Tí TJ TJ TJ TJ TJ TJ TJ TJ TJ O COin cm 10 OrtH if m en o CN CN en <-i cmcn if n en co i~i m m ^ co o m cm co cn co rn 10 -H C nj *T m en m --h r*. m o r~ O ro en v0 m cn m i-t io (N r* ro M i0 O r> r- ci r^ < ^r m LO CO (N p* en en m o CN m m co i0 CN r- lo Oí lo io <N oí O <T 0 co -H 10 c r- en co en O O 10 cn m en ^ i0 co io \o O O cji CNr* en io f 01 en CN c en en m H HO» 10 CN o r-* co co cn if if en en cm r- t CO CN l in i0 if T if rh H ^ en co en cncn O Oí r-i (N r-h CN m en lo r^ enm fm CN CO CO r m cn 01 CMm en oo i m co m en m r^ -n ^" en p- MflO CN H O* CM C CN ^ ^ roen ro en p- m < O CO CJt m rh <f co CN CTi -H niño -h m co *r c 10 o en cn en en en in CN r^ cn r"» O ^ CN r^ ro en CM CM 10 >H m i -h en n CO r-t C ch en H O en r-t mtjif m en r^- r*- i0 m r- if m co m o en ^ cn T co m in c CN r-t r- * m m co O r-t 10 O O i cn m n* m nt o 10 Tf ih C0 r-i ^r 0» m if en c <-H CO1 en CN 10 1 CO CO 00 -H COrH co CO CN tn 10 CTi 10 ID r- < >mh CTir- cn co m in O m en en cn ^ r- m O r-t m i0 i0 CO cjl CO en *T in r- c io r* en co co 00 tt CN en o m r*- en --H ro 10 ID O eo (N T*n CO CMCN r* c r> cm(n o o n CNm i0 o o r^ r^ (N m cmr> p- en <-i»0 p P~ CO r --4 en H^O o en T if rh ^f en co r^ r- in p» co en co cn CN m CN co * f*l CMCO m r-t m in i0 en o 00 CO CM CN (N T 10 CO O 10 ^r p* cn 0 10 in m <f en O en m n* <n n if *r co cn ro o m CO <N rh en 10 co en to r*- en c CD CNJCN m m co co cn oo ^r o rn <h en co co ^" fo -H don vo en co V en m r- ' O H r- m co en o 10 <* o m i0 -h co co 10 ro rh in mon u3 t cn m r- ^r c r- io -h en CN rh CO r*- ro m co - co m en co»n r- en ntnin 01 p* CN 10 i r- o p- m v0 m r*- r-i O CO CO P- in co p~ cn id un iû t in r O m CN 10 rh co in r*- en if m m J T <H H O CN m H en f H ^r en en en m r^ cn m icnencicnmcororhror*! HtvjHHConHniruDinrnHHH^nH h O O O OOHHHrlHHH r-{ r-t r-i CN o o o o o o o o ooooooooo o o o ) O O O o o o O i i I J.4.J.J.J. TJ TJ Tî TJ TJ O TJ TJ TJ TJ TJ TJ TJ T) TJ TJ ' TJ TJ TJ J TJ TJ TJ TJ TJ TJ TJ en rn en i0 en m r^ co ko co co en r^ t m m cm ) T CN rh CM (N co in en en m p- m -i en r CO CO i0 o o ^r m o ' OO«) - m r-h en CO CD rh CN co co O 10 o en r i en r co en en r-t r- ' CO <D CN CN 00 rh 10 rh r- rh ^ "40 X» i0 m (N oo m TJ 00 Tf CM 10 C0 cm m o ro en ( en o co rh m en co m en co *j O rh r- r-i *t m en co en o en 10 CN NT" ro o c [^ P- rh to m cmen in o co en ^" o en en m. i en loo't *f r~ co rh 10 co *ï r- c i0 m i 10 ^T 00 Tf o i0 O if CN rh «3 * o m en P-- t o en i0 co en ro rn P^ ro i i0 ^r r- r- r^ i o p- en lo^oo en en cm en cn m m en en r^ i* o o en en co t o p- r- en io cm10 r co en in in r-t r-l ^ r- r* ro v0 r^ cm10 10 co v0 en r rh en en en id cmco en -h en cn ^ \o r- o en i0 cn O T T m m cm i co r-t m r-~ in m m m r^ cm^0 m en 10 CO rh 00 cn m T CM T eo en r- O r- co ro cn en < o h en (N o en CN co o cn oo V$ t0 ^* ^" m y) en co en i0 m cm en cm «p- m en r- co en r-- m ñ* o co H OlOH en en en m co i o o o > m en cnoo co o \û in i0 co o T 10 r* en ro en rn 10 o CMC0 if ^f» cn en ( (N CN rh CM C0 rh en r C0 tj- m *r r^- CN en r^ co co ^* co m. m m t rh V0 00 co r- en00 10 CN CO O CN O <N CO O r^ co rn 10 cm co i co en ' i0 i0 O en ^" m CD tn rh o en cn en CN o en m i0 > r- *r co r- c I O 10 i CN OO <r id <r O CO C0 rh co m ^r m n Cn CM rh r-\ r-t r-t l I 10 CM l m rh rh Hlflf (N in vd en o if co co r- i <-{ r- m m o 10 r-t r- ( ) O C0 ( en en vo P- O O m O cmr^ oo p^ m cn io m i. en co ^T 10 P- r-t co t cm n r-\ CM 10 O O co r^ CO CM CO rh CN T ^r i0 r-i m IN CN O CO ( O <N 10 C0 F cm r- r- en en co cm co r- i0 r- co CM NT CM C *f O r-* r-\ (N 10 oo ^r en en en m co oo m co co m O CM CM C 10 CN rh CN r-* 10 en m cm m en cn o en o ro COI'OI CO CM 10 m rh r^ A1 n n o mcnl0rhlncmirini0co' irhcncn<ncnen^,ij,'ïini0i0r-cocn. jen^ini0r^ajc^ohrsicn^ini0r^cdcno'-icnrnntini0p--cocno rh-hrhrhrhrhrhrhrhrhcncmcmcmcmcncmcncncmm -i m vd r o o o c ) O o o o o TJ I TJ I i i i l l l TJ TJ TJ TJ TJ TJ TJ TJ 3TJTJTJTJTJTJTJ m cm \Û CM C0 "linrsicncn m^nr» in *f m C0 CMCO O O t en o o en lo io en r- T ID > J P- o o en in - iû r-»r oo O cm r» en m en m O C0 en r- 1 m - I CMP- 10 r- r- en «T co tnlûo oo en rh m iû CO tn P- rh 10 L v0 en \ > en rn CO it kd if vd O eo in *r if if J en CM T i co in p- C0 r co 00 if rh CMCJl 10 r. O m i0 m m m vo co m m en cm co o m r* if CO CO o t r^ to en m en id t cm tn cn rh > m r^ cm in m ^(Np> m m co O O rn r» o t-- 1 TT V0 cn *r r- cm en m eo cn O if O CN C0 10 if m rh r^ r^co «r m id if 00 CN m -t i0 \0 en if r- i0 CN rh rh r\ t0 co o en r» i m j en m i p- en ro cm T (NCfilD X) -H T if 10 o m o i vo CNo co co if CN en r- co CM rh m CM t io o - oo co O en o oo co m -h r^ O i0 \o m p~ o r» en m i (^ en (N r- <n o en en -h co en o m t CN T j r- *r o o m 10 r i p- p- m co en rj CM co m O o en t C0 CN rh -T rh o 00 m O 10 \0 r- i > ñ" en p- en CO 00 en co co 10 (NH en O 10 if en m o <-ten r- if m.-i en en 00 P- cm m en co co en co r- i o o r- <n o o en o co m h m en cm cm m t iocn-fvocncnr- JHHHHHOjrg I I I I I 1 I I co r-t r- CN l CO CO P- e r-t m *? m n t CM rh O O O O O O O O o o o o o o o o o o o o o I TJ TJ TJ TJ TJ T) TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ CO if O if -tf m en rh if C0 m in O 'D 10 p- en m o --h O 10 co P- rh m m o r- p- (MuIhNiO o o o en en m m in en CN 10 O r-t m co m cm m cn m f i0 in h if p* CM O r- o en enco if o O en en o iû m <H P- Cn rh r- m en p- O 10 O r- i0 in t o CN 10 Cn rh co m en if O CO o en en m t (N O O T P» o rh en en i0 rn en p* CN CN m o i-h en en i0 o m m ^" r- f r-t CD ^* m o rn cn en ^> m if rn m co en O CO rh if r-t P* P- 00 if -H CO if o co tn to o en in h i0 io tn 10 P* CN O C0 co en rh m (N en co m P~ en rn if en p* to i0 m* m CU rh en i0 cm oo en rh O O O P- O m CM P^ rh MOO C0 CN m en r-t O ri cn co en en en in ^" oo co en en o if o en if <f m co in o m m co r-t m if o to m t if rh co o en i0 cm r^ O 00 if 10 C0 CMcn en cm r- io r* co 10 co 10 C0 10 r-t P- *f erj o o r^ o en m m id co en 10 CO oo m en m P* if r^ m ro in if co en co i0 en co en -H en en tn h io <n co m r-- m «r cm co T rh m co 10 CO m - Hiflin r^ r-t rf m m co ^* en r> cm CM rh C0 O co co r> en r- io en rn r-t m co o cm co en en en co en 1 CO tc0 CMen co r- oo»o p- en en ñt co CO 10 en (M CM CMo cm en cmñ* en r-t cm rn if o en co en co n? [^ O r- (n ~> cm r> if CM VO(N ^" in r-h [^ en f r-t o cn rh m 10 > o 10 O CO 00 p* en m io rh en CN -H nf cn m en o t m r-t m 00 CM 10 rh yo cn en r^ co COifrHr-HCnTflOCOr ) cm en en if im* CN CN en if if in»o oo en o cmif 10 co o o o o c O O O O O o O O O rh rh r-t r-t r-t CM TJ I TJ 1 TJ I r> I I I f I I I I I I I I I I I I I I l TJ TJ TJ TJ T) TJ TJ TJ T) TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ CD 10 CO if (N i if O m vo en co tn cm v en i0 o»o m en en m vo «-n O in J COiDf co p- P- en m m 10 CO if 10 rh cm p- co m \o en \û m CM O 00 - en en if en en to en to io rh p» IDIOOT OHifin in in if c co P- P- ñ* m m oo i en o p"* co vo co m en cm en if m m -h r O P- o m p- rh O r- O rh r- co io en co m m vo rn o m 1 if CO 10 cn o p- r- oo co o r~ C0 CM if 00 p- en en r- cm in en m if i en cmcm ( C0 en cn o m <n en i0 en tn p- p~ o P- en en ro en en r m o ^0 r- o rh if cm tn r- r^ id p- in en.-h 10 p^ en en oo p- cm r- h o r- i* r- if (N p* rn eo en <ï cn p» if co o o e o m en ^ cn p- cm en m o oo o 10 rh o cm r-t rh cm co r oo o o if r-t Tf en p* co cm en in rh cm co 00 O ^ lo o en co m m e if O cm ^" 10 p^ i0 co p- o en en ^" n io r-t en en 10 CN o en m r m m o 00 co o en m en p^ m ij* m lo o enif m 10» ^f r^ cn m rn en en io co r-t o vo en cm cmm cm en en co r-t m en r i0 o m if 10 p- HVtMl/IH O 10 if CM-n tt o en en en r-t if en e if en en r-t r-t en cn en m o m en ^" cm m cn cm lo en rn en CO if P- r en co en p* en en ^ o en en rh m p- m co lo en m cm p- if en en co io»r p- cn co en p- m <3> o lo en co en C0 en if c P~ CN CN if m if rh if m en if i O en io co if n m m m > r- rh o cm cm en en rn lo p- en if p- en CM -n en p- cm h n o cm 1 in if m cm en en cm o p- m en en m en m en \o p* p~ r p- m cm t 00 m rh en en co i0 cm p~ o en en en m p- m cn en p* 0 m en en r- p- O m rh en <-t cm en r-t io if p- p~ m co m O cm u co en en c r-t 10 CN rn cm en cn en p* en <-irh en m ' p* o i en m en cm iirihi'cohhhmnhfnininnh. O O o O O o c o o O OOOOOOOOOOOOOOOO I I TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ CM CM O CMm en O en m r-h if co m r-t P- if CM CN ^" O F> r-t m 10 CO en en ^ i0 «m p- O rh r-t 00 r-t m io en io m r- cm m rn m cm en cm r-t m if P- in en m en i0 co <n en cm CM rh p- o m m i0 en co en en lo o cm ^ tn n? cm en ^" CO CN rh 00 rh 10 m f o m eo cn cm if en m m. i en en if en io m P- 10 O CM io r*- cmr-t tn o en if ^ oo 10 o m in en m en en o en co en i0 h C0 i en m rh p* 00 CM P* r-t <f en lo en oo cn <f io -aen en io io co en CMr* o en cm to ^ in en en t ^ o if p- m cm o o CM CM en o H [S en in en co cm m 10 O P* r-t o if en CO (N 10 cm -n m m m ^> rm C0 ij o r~ cn o en <j o co o cn r- o CM O CN m n* cm m if r- en 10 i0 O m cm O rh ^* co r*- en m cm co iû oo en en cm m en rn m o i en. rh rh m co o o en o -h eo r-t o to co co p*. en co co o en m if en en MOH en o r- co p* o m cn if en co en cm O P- CM 00 m en _ h m ^coo m Ni* 10 rh if en cmh- en io en en tn p* m m «HCN CN o vo en \o p- en p^ o m en en ^o en iio h rn m* en o co r-t en co p- if oo o "* "» P~ P* r-t rh en io p* f m ^* CJl rh oo en m p- co m if en m cm rn o en cn en m O 10 P- cn cm r- cm m en o co co r-t m P~ CO if r- cm if m rn m r-t r-t 00 r-t -tf 10 «n cm en en CO CMp- p-- en p~ o en r- if o r- O lo 00 r-t in P- C0 p- r-t m o o o co oo p* P^ n m en if m if co O en \0 en co o co m <r CM 10 CM 00 O r-, O h en loorji 00 CM CM CMm co if en n* io «n en eo o *r en if CN OJ i0 co m 10 P- r-t cm rn en en t co r- p* oo m en ps en co m m r- en r-t *r en H m COifiinp- ro en 10 T r- io " o i en m n? ñ* m if m en if -n O CM if P~ O en p- rh in O m h co io m io 10 1 CO rh CM en if 10 CO ' rh cn cm c encnififininiop^co rhcmcoifiniop-cocno'

17 SUPPLEMENT S3 cn cm cn en «a- m in vo p* co en o r-t cn nt in p- co o cm m 10 co 0 m m co rn m en if o ) O o oooooooo o o o o o o O O rh CM CM (N CN CN en en cn en a1 if ^r m vo I I l III i i i i i i i i i I l l l I l I I I I I till I i i i i J TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ T) TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ T if en in cn O 10 r-t rh en en in if rh CO P* r-t rh O O m en r- en cm en eo m 1 CO o o en rh in oo co in r-t en m 10 T r-t cm rn en o tn eo m cm tn if m m if cm tn en cn io p- en co o un o o en r» r» i* en MJieo in cm en en rn cn m O rh -t o m co o oo m o if in r* r^ m m -h cn p* 10 *f cn o m O 00 rh CN VO 10 P* CM CM10 CM en 10 p- p* m oo m en m p- m if p» vo en en OOlDCDI1 1 in r-t CN 10 T CM if O r-t 10 O O ID CM P- r-t if O en p* en en r-t r-t itl co f co CN C0 en io o en O en C0 cm O. co en 10 O en co en HOlfl io cn r- tn vo p- O O CMen io eo p- o m rh 10 rh if P- vo cn p- CM CM P* r-t r-t ) 10 P* H I o CN if m rn cn O lu CMCO P* P- o cn m m m o 1 CM P- en p* en CD COco m ^ en m p- co co en t io en io m h cm en 0 <f m CN if O r-t p> m r-t r- eo if o en cn p. p- p- en co co en p- co io m en cn io p* en 1 rh CM co eo m m m cn en co if o m r-. en if en oo r-t vo p» p. io io en CO 1D rh rh if m m *r en rn rh m en J if CO 10 o m r 01 f^ m en if io m o O C0 r-t o r- en O if CMif p- io cn m en oo CM o CO in CN -cf rh i oo m P- r-t en r CM eo if co if h P- NT CN H en en m h en cn m co m en r en i CN en m oo p. eo m cm co r-t t en ID io <n \ m r-t 00 co O O o r* en in en in o m io m* m CN 10 o co co in en en cn ^ m *r m io p~ *f *=f i io en rn 10> en en en en ij r- co co io r- if if co 10 p- m if nt m p- r- o io O <tf H P- en cm m if co ) m io in r-h en m en io io m m o m T en m en p- io tn o <no co eo O r-t 10 P- O en io cn en io cn p~ eo p- r o en en p- io en co O if H CMen f 00 r-t p* m r-t t I 10 0 m cn en eo r-t co en cm r-t p* in p- *f cn m rn t cm cn m mi tn p* en 10 r in m if en rn rn co oo oo en io oo cm r- O o r*- en 10 if OM^in CN rh p- o *f l en co io io» r-t O 10 cm en r-t un m id m if r-t m en co oo co m C0 if if p- p- m p- en m co en io m r-t p* io m I CO io mi CM T r-t cn p- p- 00 if 1 o en cn <t.p- to in o cmco P- o m o en rh m cn oo r-t CN r-t m CN o i0 if nt eno o cn p- (N if O o r- eo m m p- r*- p* lo if co p- o o en en m rn o en ) 1 en if Ht p~ in vo r-t 10 ro O 00 rh io cm m co p o co if m io m cmio en r-t io tn cm m m en m en m 1 if 10 en o > en o en co cn eo O tn i m p- p- en cm p* CN if P- io io enif 10 CM en m vo io to m cm en o in ioi co m m o oo oo io 10 r-t «T 10 O CN r-t m p- if 1 o cm cm co?? o if i** en P"- T P* en en t co j f if m cm oo m r- r- cn r-t p- en rh CJl P^ tn in in O O rh m cm o h m cn en r-t r-t co cm vo OrHcoOiootnm V0 T CMtn o O p* r- oo m i p- io m m OMCÍ rh cm o oo p~ p- en cm m o en ^ jcmencmcnrhr^tnrhinrhicnhtmtnit^enenrhr^enencminoorheniûcmco J O O o o o o o o o o o O o o o O o ( o o o O O O o o o O O O i J TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ ' TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ T O f CMco in co m m oo cm io en tn C0 rh r-t 00 t 1 10 o m cm10 10 CMif io en if 10 P O if O 10 O h io en co m cm oo 10 CN P- p- r» en m r-- en if en eo en en o r- 00 rh CMo o o rh en en en p- p~ co cn lof IN Cn rh r-t r-t 00 CMp- p- co if en io 10 CN p- m en eo p* if en P- -tf 10 m CN rh CM 10 if ih en m p- <f - co en oo m r^ if r-t r-t 10 oo p- m CN r f CN C0 10 cn en p* m io O CM if if cm m co p- en r-t tn j r*- cn cn to rh p- cm m cmio co o en io cn 10 en CO if o CM O C0 p- if m rh T C0 en p- cn HOMOCO J P- VO CN CN O -H 0 cm cn oo en cn p* io m p- io if CN io m io m io -t p- en CM O CN o vo en <n cn en if tn r n o r-t O r-t 00 en oo cmrh m CMif CM co m vo en rh o en en p- cm r* en H CM CMCfO en rh m m rn j en if rn o o O P* P- rh ^MD P- rh m io io rf m o p» O r-t 00 P- rh CD 00 rh if <N cn en vo.h o vo r co eo m if C0 10 VO P- CMm io r- <í MN en f 10 en h <r 10 O en if cn m P- rh 00 en m m m r* p* o vo j ^ r-t m cn ^ m 1 P- o en m en en m oo 10 CN o m r-h 10 if CN O mm^ r- cm en O -H O r- ^ oo cn co J r-t cm ^ nt o 10 en ^f cn vo en en en r-t if en m rn r* en O cn cmrh IO 10 m if cm if m H CM <D 10 io m en ri ID eo rh 10 rh ^ vo 0 O co if e if OO VOm p- vo CM h m co cm m oo en rn en en J co 10 ^f rn io p- en vo h en p- en o o o p- CM ) U0 r-t r-t r-t r-t 0> O O rh oo r- en en p- o vo cm en if f t en en if m io h r-- en M* r-t CN co cn m rn lû io en co en 10 r- COif en en oo p* en o <r en cm p* Is- CN P* i en m en en - co en m m if en r» m en p* 1 o O ip m p- m o -h -i r- en 10 cm o 10 if rh m io m m oo j i en o m p* en oo en o r* o VO rh 10 O m p» CM CN CO 10 rh O 10 CMoo m m CN rh o p- ih nt" in co g f MN cm en co co en if r-t if VO p- p m cn m io en o p» o cm en O CO 10 CM CO rh m p- io o m n? m cn > Tf IO if en r-t co p- m eo en rh co eo en en 1 m O if f m H in o r- en if o m oo CO O CN io if t> o 'T 3 o en o io oo co m >* r> r-t rh CO p- en m r- p- -i 10 o co en m i en < m 10 CM P^ en if en CO rh r rh o r> 10 r-t C0 nt P-Pif io m io if if 00 r-t P- rh en co rn -rf O CM rh CM cn o r-t m rh cm m en en en en en r- m mu 10 co en r co en en if vo p- vo en if rh P- eo if cmio r^ co Tf en j m en cn rh CM P^ o en en o en if ^, «n m en m co cmo CN O CM rh m f> p- m m rh r-t ^ CM p- rn r*- en h en rn oo cm p* en o i co en m cm m oo co r-t 00 tû NT" en m in en p«p- en p- o m if m o o en co en n* HHinif) cm o o O CM 10 o o CNiT 10 CO m eo rh if en f eo en co i OOHCSif 1lNlno^HNn^ln^o^cJiHHHHHr^lr^lNM^^nm1"^^lnlnlOlO^a]c0WHHHHH r-t cm rn -r m io ^ajc10hn^^^ln<o^coljloh^rovull0^cd*ohc^i^nt'lnl0^co!jlo rhrhrhhrhihrhrhrhrhcn<ncncncncnincncncn cnencnmcncntntncnmn? rh rh CM CN en m if tt m io p- co en o c >ifi0cooenifi0cno o o o o O O O O 0 o O o o O O O O rh r I CN t J CN ( I I 1111 i i I I I i i i i i I I I I I i I TJ TJ TJ TJ TJ TJ TJ TJ T) TJ TJ TJ TJ TJ TJ TJ TJ TJ i TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ i ^ en en co m 1 10 n? T O p- m co io -h o en if e 10 -T O en m in m O m O m rn m o co t en m o ^r rh rh CM O io cm io en rh P- O -H 1 r co en en cm co o O o CO o f rf en co P* en if in CM CM 00 O io m co en -h p~ m en o eo ' en m io h o cn oo r- io nt en p- m en co I en 10 r- m O O co m 1 cm en en r- <H rn p- if e en m cn m co if en m rn o eo m p- en o ) eo en r> if 10 CM 10 en en m if co en co i co en m i CO rh (^ m rh if CN m eo CN 00 if > if rh r- ce m CN CM \0 h ^r m m tn CN CM rh co en in -H in en m -h o m -h r-t io m en m en ) P- CMO co rn r-t i co C0-5T io en r- O m io o --t c f 10 if CO rh CO O co m m o -h if VO rh \ io r* CN 10 p- en o m co m i0 io if if en en en cm r CN rh CM 10 P- CN rn m en en oo m co if t eo m if en if m co en O in O en O -H i CN to 10 e tn m m m cn P- nt rsi en nt en vo if en io ) r- -h en cn o o <f r» m m m m p- iot h h t cm en io O 10 P- Tf ^ if r- rh "" oo o 1 00 rh en cn cm ^f en rn p- m -h r- 1 en en io p- m - m <r if CM co cm m en cn r vo p* en m > m io en io o m V i> co CNen co cmp- co io en rh c CM CM rh io id to eo CI m m o en j in t rn p- * o rh Tf cm p* O rh Mnn co r- vo io r- t 10 CN 10 ^" co m if cm en rh 10 CO J 10 CN o 10 rh ^ m cn if m rh o r^ if T en if tn i io cm en r- m (-» vo en ^ en if i ) co o O cn oo in en en CN if P^ 10 rh p- >h eo en m t en p- -n oo r- id vo vo rn i m en t O 00 co t m m ^" rn m en en eo P- O r- co co f c T P' if if m o en vo vo rh in m i rn fm m <f p* io m cn p- m if cn r- if rh o m p- i O CM 10 rh if CM vo co co co P- O O io en en ^ cm0 rn in io vo en -n io cn t 10 1 CM P- co m en i en p* in p» en p^ oo cmr- if o cn m io en co en o ' in cm if io p~ en r- > if r-t en io en p- en io r-- en en m en i en r- cn cm [ -h o r- p- o en -h 1 p- p» p- en if en p- o m p» m cn if H) CM if CN P- ( COHOlO co co lo ) O en o r- p- co m en p- o en t m r^ co rh co if in rh o r- O CN CN poh m en r- CD 10 r- o co vo 1 t- en rh oo in O o p- en c cm en eo 0 in co i r- co if 10 1 co m p- r-h 10 CO rh if CM eo cm rr rh CD VO rh o en m < tn o p- 1 C0 CM rh o r- co ao in en co o rh cm en m e ihp-mhcsinr-coioncrihhcor rh O O O O O O O o o o O O O O o o > o o o o o ooooooooo I I TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ ) TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ TJ rh * en en p~ if) P- Ol CO O in en f p- en 1 CN T if CM VOp- en o rn en o vo P- C0 m co r- m p- en m i if rh O CM rh O r-t P- if r- <f m rh en en vo en CO en rh co m en -h io r» if io m rh CM O co oo m -h p- en en cm if en h 10 (-. co r- en m en P- 10 en cm o 10 CN P- o en oo vo in if rh rh if if vo tn i* to en o co en ^r o en m i0 H en o cn o m <. _ (N en to m o en vo vo r- ^f m P- CM 10 cf cm en CO r-h CO CM co en CO CN if m r- OPT O en m en vo CM VO CO en vo en ^ 10 rh en o r-h m 1 CD i CO rh if o co O en m en - co eo r> en ^ cm10 VO CMrH m m VO in rh m if rh 1 m cn m rn m o r» r- *r rh CMcm en en rn -H r-- p^ in if p- cn cn 10 if o ^f rh r-t rn r CN r r- en en en in en nt en co m en CM O rh o vo cm en o t en i en co cmm in co CN VO CMcn o T if vo en O 10 -H r-t CO O m m rn O O 1 CD -H to if m co c en cm vo o rh co 10 m rh r-t m p- r-- en CD 10 en ^r rn en m vo oo o co en cn cn if CN p- m in m if cmv0 O en 00 o co io en «a* co cn CN CN p- if oo vo O ro eo cmm if cn vo en 10 if en en r- p- en O en rh H P- rh 10 CN ^ m io p- vo H 10 oo to en rf en en en if en m <* m n o H en cn p- m vo n* en en O vo en if ^ eo rn O ^" P- CM 10 «h m m O CN CMco cm en en en m io vo vo m cm cmv0 en cn cn m if en en vo u-l CM vo if en 10 O rh p- en m p- CM O en m en en cm p- 10 O V0 lo P- o if rh o rh CN i O VOen en m P-lOP O en m C0 rh O CN O nt C0 CM C0 loh cm p- Pinp CD if rn en if oo oo o o m cm ri rr VO 00 p- ^lat if en if if en cmo m en en o p- m co en co vo if m co P- en o co co CN VO rh Tf Tf co O f 00 rh CO O h en o "i rn < i en m m vo co COOP en m cmcn oo en rh m vo vo cm en oo tn rh <» en vo vo en m in cm tn i io m p- rh VO O if r^ co en if -n co p~ en i io co CN 10 P- cm tn i ) CM 10 m.. t j en - rh m o im n* m rh 10 P- io en P- 10 CM1 p- o eo en m co m io p- cm en V0 O ' I CM p- rh io en P- rh O, < en co en v0 o m en en 00 O if vo n* vo p- o rh vo p» m h *f [^ < ) P- O rh O rh co m p. -.-h o _ en en m en CN en m o m m en oo h (Jim <T CM CM^" co en 0 o ' o en O T rh rhvoinorhcnorhif 10 if en r-t r- tn ^" in oo -t p- O CM f 10 CO rh 1 p- t ) r- o m en nt en i o i itncn^rmp-corhr H rh CM C lencnroififirininvop-p-coenrhrhr if rh