Summation of Series and Gaussian Quadratures

Transcription

1 Summation of Series Gaussian Quadratures GRADIMIR V. MILOVANOVIĆ Dedicated to Walter Gautschi on the occasion of his 65th birthday Abstract. In 985, Gautschi the author constructed Gaussian quadrature formulae on (, + ) involving Einstein Fermi functions as weights applied then to the summation of slowly convergent series which can be represented in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. A problem that may arise in this procedure is the determination of the respective inverse Laplace transform. For the class of slowly convergent series + k= (±) k a k with a k = k ν R(k), where < ν R(s) is a rational function, Gautschi recently solved this problem. In the present paper, using complex integration constructing Gauss-Christoffel quadratures on (, + ) with respect to the weight functions w (t) = / cosh 2 t w 2 (t) = sinh t/ cosh 2 t, we reduce the series f(k) ( ) k f(k) to weighted integrals of f involving weights w w 2, respectively. We illustrate this method with a few numerical examples.. INTRODUCTION We consider the summation of series of the type S m = T m = a k (.) ( ) k a k, (.2) 99 Mathematics Subject Classification. Primary 4A25; Secondary 3E2, 65D32, 33C45.

2 where m Z. Methods of summation can be found, for example, in the books of Henrici [], Lindelöf [2], Mitrinović Kečkić [3]. In a joint paper with Gautschi [9] we considered the construction of Gaussian quadrature formulas on (, + ) with respect to the weight functions w(t) = ε(t) = t e t (Einstein s function) w(t) = ϕ(t) = e t (Fermi s function) + showed that these formulae can be applied to sum slowly convergent series of the form T S whose general term is expressible in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. Namely, if a k = F (k), where we have F (s) = S = T = k= k= Also, for a k = F (k), we have S = k= e st f(t) dt, Re s, F (k) = f(t)ε(t) dt ( ) k F (k) = ( ) k F (k) = f(t)tϕ(t) dt. f(t)ϕ(t) dt. If the series T S are slowly convergent the respective function f on the right of the equalities above is smooth, then low-order Gaussian quadrature n g(t)w(t) dt = λ ν g(τ ν ) + R n (g), ν= The functions t ε(t) t ϕ(t) arise in solid state physics. Integrals with respect to the measure dλ(t) = ε(t) r dt, r = r = 2, are widely used in phonon statistics lattice specific heats occur also in the study of radiative recombination processes. Similarly, integrals with ϕ(t) are encountered in the dynamics of electrons in metals. 2

3 with w(t) = ε(t) w(t) = ϕ(t), applied to the integrals on the right, provides a possible summation procedure. Numerical examples show fast convergence of this procedure (see [9, 4]). In the sequel we refer to this procedure as the Laplace transform method. A problem which arises with this procedure is the determination of the original function f for a given series. For some other applications see Gautschi [6] [7]. In [6], Gautschi treated the case when a k = k ν R(k), where < ν R( ) is a rational function R(s) = P (s)/q(s), with P, Q real polynomials of degrees deg P deg Q. By interpreting the terms in T S again as Laplace transforms at integer values, Gautschi expressed the sum of the series as a weighted integral over R + of certain special functions related to the incomplete gamma function. The weighting involves the product of a fractional power either Einstein s function ε(t) (for T ) or Fermi s function ϕ(t) (for S ). The case ν = of purely rational series complements some traditional techniques of summations via quadratures (cf. [, 7.2II]). In particular, Gautschi [6] analyzed examples with a k = k /2 /(k + a) m, where Re a m. The series T with a = m = appeared in a study of spirals given by Davis [] (see also Gautschi [8]). In this paper we give an alternative summation/integration procedure for the series (.) (.2) when a k = f(k), where z f(z) is an analytic function in the region {z C Re z α, m < α < m}. (.3) Our method requires the indefinite integral F of f chosen so as to satisfy certain decay properties ((C) (C3) below). Using contour integration over a rectangle in the complex plane we are able to reduce T m S m to a problem of quadrature on (, + ) with respect to the weight functions respectively. w (t) = cosh 2 t w 2 (t) = sinh t cosh 2 t, (.4) The contour integration is discussed in 2. The generation of the recursion coefficients in the three-term recurrence relation for the (monic) orthogonal polynomials π k ( ) = π k ( ; w p ), k =,,..., with respect to the weight function w p (t), p =, 2, is discussed in 3. Numerical examples are presented in 4. 3

4 Figure 2. The contour of integration 2. PRELIMINARIES Assume that f g are analytic functions in a certain domain D of the complex plane with singularities a, a 2,... b, b 2,..., respectively, in a region G = int Γ ( D), where Γ is a closed contour. Then by Cauchy s residual theorem, we have 2πi Γ f(z)g(z) dz = ν ( ) Res f(z)g(z) + z=a ν ν ( ) Res f(z)g(z). (2.) z=b ν Let { G = z C α Re z β, Im z δ }, π where m < α < m, n < β < n + (m, n Z, m n), Γ = G (see Figure 2.), g(z) = π/ tan πz. Then from (2.) it immediately follows that (cf. [3, p. 22]) T m,n = n ν=m f(ν) = 2πi Γ π f(z) tan πz dz ν Similarly, for g(z) = π/ sin πz we have n S m,n = ( ) ν f(ν) = π f(z) ν=m 2πi Γ sin πz dz ν 4 ( Res f(z) z=a ν ( Res f(z) z=a ν π tan πz ). π sin πz ).

5 For a holomorphic function z f(z) in G, the last formulas become T m,n = 2πi Γ π f(z) tan πz dz S m,n = 2πi After integration by parts, formulas (2.2) reduce to Γ π f(z) dz. (2.2) sin πz T m,n = ( π ) 2F (z) dz (2.3) 2πi Γ sin πz S m,n = ( π ) 2 cos πz F (z) dz, (2.4) 2πi Γ sin πz where z F (z) is an integral of z f(z). Assume now the following conditions for the function z F (z) (cf. [2, p. 57]): (C) F is a holomorphic function in the region (.3); (C2) (C3) lim t + e c t F (x + it/π) =, uniformly for x α; e c t F (x + it/π) dt =, lim x + where c = 2 or c =, when we consider T m,n or S n,m, respectively. Set α = m /2 β = n + /2. On the lines z = x ± i(δ/π), we have that π 2iπ = = sin πz e iπz e iπz 2πe δ e 2δ e ±i2πx 2πe δ e 2δ also cos πz = 2 eδ + e 2δ e ±i2πx 2 eδ ( + e 2δ ). Therefore, under condition (C2), the integrals on the lines z = x ± i(δ/π), α x β, 2πi β±i(δ/π) α±i(δ/π) ( π ) 2F (z) dz, sin πz 2πi tend to zero when δ +. For z = β + iy we have β±i(δ/π) α±i(δ/π) ( π ) 2 cos πz F (z) dz, sin πz sin πz = ( ) n cosh πy, cos πz = i( ) n+ sinh πy, 5

6 π sin πz 2 = π 2 cosh 2 πy = 4π 2 (e πy + e πy ) 2 4π2 e 2π y, π 2 cos πz 2π 2 e π y, sin πz so that β+i(δ/π) ( π ) 2F δ (z) dz 2 e 2 t F (β + it/π) dt 2πi β i(δ/π) sin πz δ 2πi β+i(δ/π) β i(δ/π) ( π ) 2 cos πz F (z) dz sin πz δ δ e t F (β + it/π) dt. When δ + n + (i.e., β + ), because of (C3), the previous integrals tend to zero. Thus, when δ + n +, the integrals in (2.3) (2.4) over Γ reduce to integrals along the line z = α + iy ( < y < + ), so that i.e., T m = T m, = 2πi S m = S m, = 2πi α+i α i Equality (2.5) can be reduced to T m = 2 T m = where w is defined in (.4) Similarly, (2.7) reduces to α+i α i ( π ) 2F (z) dz (2.5) sin πz ( π ) 2 cos πz F (z) dz. (2.6) sin πz cosh 2 F (α + it/π) dt, t Φ (α, t/π) w (t) dt, (2.7) Φ(x, y) = [F (x + iy) + F (x iy)]. (2.8) 2 S m = Ψ (α, t/π) w 2 (t) dt, (2.9) 6

7 where w 2 is also defined in (.4) Ψ(x, y) = ( )m 2i [F (x + iy) F (x iy)]. (2.) Formulas (2.7) (2.9) suggest to apply Gaussian quadrature to the integrals on the right, using the weight functions w w 2, respectively. The required orthogonal polynomials can be computed using the discretized Stieltjes procedure (see Gautschi [3, 2.2], [4 5]). Instead of reducing integration to the positive half-line, one might keep integration over the full real line note that S m = T m = Φ (α, t/(2π)) e t ( + e t dt (2.) ) 2 e t Ψ (α, t/(2π)) sinh(t/2) ( + e t dt. (2.2) ) 2 Here, the weight function is t w(t) = e t /( + e t ) 2, the logistic weight, for which the recursion coefficients for the respective orthogonal polynomials are explicitly known 2 : α k =, β =, β k = k4 π 2 4k 2, k =, 2,.... Thus, no procedure is required to generate the recursion coefficients. Some comments on the convergence of the corresponding Gaussian quadrature will be given in GENERATION OF THE RECURSION COEFFICIENTS Let π k ( ) = π k ( ; w p ), k =,,..., be the (monic) polynomials orthogonal with respect to the weight function t w p (t), p =, 2, on (, + ), where w (t) = cosh 2 t 2 The referee pointed out this fact. w 2 (t) = sinh t cosh 2 t. (3.) 7

8 They satisfy the three-term recurrence relation where π k+ (t) = (t α k )π k (t) β k π k (t), k =,, 2,..., α k = α k (w p ), β k = β k (w p ) π (t) =, π (t) =, ( ) β (w p ) = w p (t) dt. Knowing the first n of these coefficients α k, β k, k =,,..., n, one can easily obtain the k-point Gaussian quadrature formula k g(t)w p (t) dt = λ ν g(τ ν ) + R k,p (g), R k,p (P 2k ), (3.2) ν= for any k with k n. The nodes τ ν = τ (k) ν,p, indeed, are the eigenvalues of the symmetric tridiagonal Jacobi matrix J k (w p ) = α β β α β2 β2 α βk βk α k while the weights λ ν = λ (k) ν,p are given by λ ν = β vν, 2 in terms of the first components v ν, of the corresponding normalized eigenvectors (cf. Gautschi [2, 5.] Golub Welsch []). In order to construct the recursion coefficients we use the discretized Stieltjes procedure as in [9], with the discretization based on the Gauss- Laguerre quadrature rule,, p(t)w (t) dt = = p(t/2) 2 ( + e t ) 2 e t dt N λ L k p(τ L 2 k /2) k= ( + e τ k L ) 2 8

9 p(t)w 2 (t) dt = p(t) 2( e 2t ) ( + e 2t ) 2 e t dt = N k= λ L k p(τ L k ) 2( e 2τ L k ) ( + e 2τ L k ) 2, where p P. Here, τk L λl k are the parameters of the N-point Gauss- Laguerre quadrature formula. The first 4 recursion coefficients (n = 4) can be obtained accurately to 3 decimal digits with N = 52 for w N = 72 for w 2. We used the MICROVAX 34 in Q-arithmetic (machine precision ). The same results were obtained also by a discretization procedure based on the composite Fejér quadrature rule, decomposing the interval of integration into four subintervals, [, + ] = [, ] [, ] [, 5] [5, + ] using N = 28 points on each subinterval in both cases. We tabulate the results to 29 decimals in Table 2 of the Appendix. Using (2.7) (2.) (3.2) we can state the following results: Theorem 3.. Let F be an integral of f such that the conditions (C), (C2), (C3) are satisfied with c = 2. If λ ν = λ (n) ν, τ ν = τ (n) ν, are the parameters of the n-point Gaussian quadrature (3.2) with weight function w, then T m = f(k) = where Φ is defined by (2.8). n ν= λ (n) ν, Φ(m 2, π τ (n) ν, ) + R n,(φ), Theorem 2. Let F be an integral of f such that the conditions (C), (C2), (C3) are satisfied with c =. If λ ν = λ (n) ν,2 τ ν = τ (n) ν,2 are the parameters of the n-point Gaussian quadrature (3.2) with weight function w 2, then S m = ( ) k f(k) = where Ψ is defined by (2.). n ν= λ (n) ν,2 Ψ(m 2, π τ (n) ν,2 ) + R n,2(ψ), Remark 3.. If λ ν = λ ν (n) τ ν = τ ν (n) are the parameters of the n-point Gaussian quadrature with respect to the logistic weight w, then according 9

10 to (2.) (2.2), S m = T m = f(k) ( ) k f(k) = n ν= n ν= λ ν Φ(m 2, 2π τ ν) (3.3) λ ν Ψ(m 2, 2π τ ν) sinh(τ ν /2), (3.4) where Φ Ψ are defined by (2.8) (2.), respectively. In Example 4. below, we will also consider these formulas. 4. NUMERICAL EXAMPLES In this section we illustrate our method using a few examples from [9] [6]. All computations were done in Q-arithmetic on the MICROVAX 34 computer. Example 4.. Consider T = k= (k + ) 2 = π2 + 6 S ( ) k = (k + ) 2 = π2 2. k= Here, f(z) = (z + ) 2, F (z) = (z + ), the integration constant being zero on account of the condition (C3). Thus, Φ(x, y) = Re Ψ(x, y) = Im z + = z + = x + (x + ) 2 + y 2 y (x + ) 2 + y 2. Table 4. shows the n-point approximations T (n) S (n) to T S, respectively, together with the relative errors r n (T ) r n (S ), for n = 5(5)4. In each entry the first digit in error is underlined. (Numbers in parentheses indicate decimal exponents.) The corresponding relative errors in the Laplace transform method [9] applied to T are given in Table 4.4. The recursion coefficients α k, β k for orthogonal polynomials π k ( ; ε) were calculated also with 3 correct decimal digits.

11 As we can see, for smaller values of n ( 5) we obtained better results than in the Laplace transform method. Furthermore, these results can be significantly improved if we apply this method to sum the series T m, m >. That is, we use T = m k= (k + ) 2 + T m, T m = (k + ) 2. (4.) Then, for m = 2()5 we obtain results whose relative errors are presented in Table 4.2. Also, in Table 4.3 we present the corresponding results for the sum S expressed in a similar way. Table 4. Gaussian approximation of the sums T S relative errors n T (n) r n (T ) S (n) r n (S ) ( 7) ( 4) ( ) ( 7) ( 3) ( 9) ( 5) ( ) ( 8) ( 2) ( 9) ( 3) ( 2) ( 4) ( 22) ( 5) Table 4.2 Relative errors in Gaussian approximation of the sum T expressed in the form (4.) for m = 2()5 n m = 2 m = 3 m = 4 m = ( 9).9( ) 8.6( 2) 3.7( 3).( 3).7( 6) 7.9( 8) 2.( 9) 5 3.8( 7) 3.7( 2).( 22) 3.8( 25) 2 4.( 2).2( 24).9( 27) 2.3( 29) 25.( 22) 2.( 27) 2.6( 3) 2.5( 33) 3.4( 25).( 3) 2.2( 33) ( 27) 2.4( 32) 4 3.6( 3) The rapidly increasing of convergence of the summation process as m increases in due to the poles ±i(m + 2 )π of Φ(m 2, t π ) moving away from

12 the real line. Table 4.3 Relative errors in Gaussian approximation of the sum S = m ( ) k (k + ) + S m for m = 2()5 k= n m = 2 m = 3 m = 4 m = 5 5.9( 6) 2.2( 7).5( 8) 4.5( ).9( 9) 2.3( 2).( 2).( 4) 5 8.( 3).9( 5) 3.2( 6) 9.2( 8) 2 6.6( 4).( 6) 6.2( 9) 7.6( 2) ( 6) 6.8( 9) 2.5( 2).3( 23) 3 2.4( 8) 9.7( 2).2( 24).3( 26) 35.( 9) 4.3( 23).3( 25) 2.( 28) 4 5.6( 2).3( 24).( 27) 9.8( 3) Table 4.4 Relative errors in Gaussian approximation of the sum T using the Laplace transform method for m = ()3 n m = m = 2 m = ( 4) 8.4( 3) 3.( 2).( 8).8( 5) 3.8( 4) 5 3.2( 3) 2.8( 8) 3.7( 6) 2 8.( 8) 3.9( ) 3.( 8) 25.8( 22) 5.( 4) 2.5( ) 3 3.9( 27) 6.3( 7).9( 2) ( 32) 7.6( 2).4( 4) 4 4.6( 33) 8.8( 23).( 6) It is interesting to note that a similar approach with the Laplace transform method does not lead to acceleration of convergence. For example, in the case of (4.), we have that T m = k= + (k + m) 2 = ε(t)e mt dt. Then, applying Gaussian quadrature to the integral on the right, using w(t) = ε(t) as a weight function on (, + ), we can obtain approximations for the sum T for different values of n m. The corresponding 2

13 relative errors for n = 5(5)4 m = ()3 are presented in Table 4.4. As we can see, the convergence of the process (as m increases) slows down considerably. The reason for this is the behavior of the function t e mt, which tends to a discontinuous function when m +. On the other h, the function is entire, which explains the ultimately much better results in Table 4.4 when m =. Table 4.5 Relative errors in Gaussian approximation of the sum T S with respect to the logistic weight for m = ()5 n m = m = 2 m = 3 m = 4 m = 5 5 T 4.7( 5) 5.2( 7).9( 8).5( 9).8( ) S.( 3).( 3) 8.2( 4) 6.3( 4) 4.8( 4) T.( 6).2( 9) 6.2( 2) 8.( 4) 2.( 5) S 4.( 6).3( 7).3( 7).( 7).( 7) 5 T.( 7) 2.8( ) 3.4( 4).2( 6) 8.9( 9) S 4.( 7).2( ).7( ).6( ).5( ) 2 T 2.( 8).8( 2) 7.5( 6) 9.4( 9) 2.7( 2) S 7.5( 8) 6.5( 2) 5.( 5) 2.2( 5) 2.( 5) 25 T 5.5( 9) 2.( 3) 3.7( 7) 2.( 2) 2.6( 23) S 2.( 8) 7.5( 3).4( 6) 3.8( 9) 2.9( 9) 3 T.9( 9) 3.5( 4) 3.( 8) 8.6( 22) 5.6( 25) S 7.( 9).3( 3).( 7) 3.( 2) 4.3( 23) 35 T 7.7( ) 7.7( 5) 3.8( 9) 5.8( 23) 2.( 26) S 2.8( 9) 2.8( 4).4( 8) 2.( 22) 7.( 26) 4 T 3.5( ) 2.( 5) 6.( 2) 5.5( 24).2( 27) S.3( 9) 7.5( 5) 2.2( 9) 2.( 23) 4.4( 27) It is interesting to mention that Gaussian quadrature over the full real line with respect to the logistic function (cf. (3.3) (3.4)) converges considerably more slowly than shown in Tables for one-sided integration, even though the poles of the integr have a distance twice as large from the real line. The reason, probably, is that these poles are now centered over the integral of integration, whereas in (2.7) (2.9) they are located over the left endpoint of the interval. Numerical results for n = 5(5)4 m = ()5 are given in Table 4.5. Example 4.2. The application of the Laplace transform method to the 3

14 series k= (k )k 3 exp( /k) = (4.2) leads to an itegration of the Bessel function t J (2 t). However, we work here with the exponential function F (z) = e /z /z, i.e., Φ(x, y) = r 2 e x/r2( x cos y r 2 + y sin y r 2 ), r 2 = x 2 + y 2. As to accuracy, a similar situation prevails as in the previous example. Table 4.6 shows the relative errors in Gaussian approximations for n = 2(4)8 m = ()3. Table 4.6 Relative errors in Gaussian approximation of the sum (4.2) n m = m = 2 m = ( 3).2( 5) 2.( 8) 6.3( 4) 3.7( 8).2( ).8( 5) 3.7( ) 9.9( 4) 4.2( 6).2( 2).2( 6) 8.3( 7) 8.5( 5) 6.6( 9) Example 4.3. Consider now T (a) = k= k(k + a). (4.3) This series with a = appeared in a study of spirals (see Davis []) defines the Theodorus constant. Gautschi [8, p. 69] mentioned that the first terms of the series T () gives the result , i.e., T ().86 (only 3-digit accuracy). Using the method of Laplace transform, Gautschi (see [6, Example 5.] [8]) calculated (4.3) for a =.5,., 2., 4., 8., 6., 32. As a increases, the convergence of the Gauss quadrature formula slows down considerably. For example, when a = 8. we have results with relative errors presented in Table 4.7. In order to achieve better accuracy, when a is large, Gautschi [6] used stratified summation by letting k = λ+κa summing over all κ for 4

15 λ =, 2,..., a, where a = a denotes the largest integer a (a = a +a, a, a < ). Table 4.7 Relative errors in the method of Laplace transform for the series (4.3) with a = 8. n = 5 n = n = 5 n = 2 n = 25 n = 3 n = 35 n = 4.4( ) 2.3( 2).5( 3).9( 4) 2.5( 5) 2.( 6) 2.5( 7) 2.6( 8) Here, we can apply directly our method to (4.3) with F (z) = 2 z (arctan a a π ), 2 where the integration constant is taken so that F ( ) =. For computing the arctan function in the complex plane (z 2 ) we use the formula arctan z = 2 arg(u + iv) + i 4 log x2 + (y + ) 2 x 2 + (y ) 2, where z = x + iy, u = x 2 y 2, v = 2x. Table 4.8 Relative errors in Gaussian approximation of the sum (4.4) for m = 4 n a =.5 a =. a = 2. a = ( ) 8.4( 2) 4.5( 2) 2.6( 2) 6.8( 8) 4.4( 8) 2.2( 8).2( 8) 5 5.4( 22) 2.7( 22).6( 22).( 22) 2.2( 25) 5.9( 26) 3.3( 26) 2.( 26) 25.( 28) 5.2( 29) 3.( 29).9( 29) 3.( 3) 5.7( 32) 3.3( 32) 2.( 32) n a = 8. a = 6. a = 32. a = ( 2).( 2) 7.6( 3) 5.2( 3) 7.7( 9) 5.( 9) 3.4( 9) 2.4( 9) 5 6.7( 23) 4.5( 23) 3.( 23) 2.( 23) 2.3( 26) 8.7( 27) 5.9( 27) 4.( 27) 25.2( 29) 8.( 3) 5.5( 3) 3.8( 3) 3.3( 32) 9.( 33) 6.2( 33) 3.6( 33) 5

16 As before, we can represent (4.3) in the form T (a) = m k= k(k + a) + T m (a), T m (a) = k(k + a), (4.4) then use Gaussian quadrature formula to calculate T m (a). Relative errors in approximations for T (a), when m = 4 a = p s, s = ()7, where p =.5 p s+ = 2p s, are displayed in Table 4.8. As we can see from Table 4.8, the method presented is very efficient. Moreover, its convergence is slightly faster if the parameter a is larger. The exact sums T (a) (to 3 significant digits), as determined by Gaussian quadrature, are, respectively, T (.5) = , T () = , T (2) = , T (4) = , T (8) = , T (6) = , T (32) = , T (64) = Remark 4.. The hyperbolic sine can be included as a factor in the integr of S m so that we use only the first weight function w. Some further investigations including a such approach will be given elsewhere. ACKNOWLEDGMENT I would like to thank the referee for his valuable suggestions improving the presentation of the paper Remark 4.. APPENDIX Recursion coefficients α k, β k for the (monic) polynomials π k ( ; w ) π k ( ; w 2 ) orthogonal on [, + ] with respect to the weight functions w (t) = / cosh 2 t w 2 (t) = sinh t/ cosh 2 t. 6

17 Table Recursion coefficients for the polynomials {π k ( ; w )} k alpha(k) beta(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 D D D D D D D D D D D D D D D D D D+2 7

18 Table 2 Recursion coefficients for the polynomials {π k ( ; w 2 )} k alpha(k) beta(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 D D D D D D D D D D D D D D D D D D+3 8

19 REFERENCES. P.J. Davis, Spirals: from Theodorus to Chaos, A.K. Peters, Wellesley, MA, W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, In: E.B. Christoffel The Influence of his Work in Mathematics the Physical Sciences (P.L. Butzer F. Fehér, eds.), pp Birkhäuser Verlag, Basel, W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Stat. Comput. 3 (982), W. Gautschi, Computational aspects of orthogonal polynomials, In: Orthogonal Polynomials Theory Practice (P. Nevai, ed.), pp NATO ASI Series, Series C: Mathematical Physical Sciences, Vol. 294, Kluwer, Dordrecht, W. Gautschi, Computational problems applications of orthogonal polynomials, In: Orthogonal Polynomials Their Applications (C. Brezinski et al., eds.), pp IMACS Annals on Computing Applied Mathematics, Vol. 9, Baltzer, Basel, W. Gautschi, A class of slowly convergent series their summation by Gaussian quadrature, Math. Comp. 57 (99), W. Gautschi, On certain slowly convergent series occurring in plate contact problems, Math. Comp. 57 (99), W. Gautschi, The Spiral of Theodorus, special functions, numerical analysis, Suplement A in []. 9. W. Gautschi G.V. Milovanović, Gaussian quadrature involving Einstein Fermi functions with an application to summation of series, Math. Comp. 44 (985), G.H. Golub J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (969), P. Henrici, Applied Computational Complex Analysis, Vol., Wiley, New York, E. Lindelöf, Le calcul des résidus, Gauthier Villars, Paris, D.S. Mitrinović J.D. Kečkić, The Cauchy Method of Residues Theory Applications, Reidel, Dordrecht, 984. Faculty of Electronic Engineering, Department of Mathematics, University of Niš, P. O. Box 73, 8 Niš, Yugoslavia 9