Evaluation of the Integral ftfu"e "\ u + x) ' du

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mathematics of computation volume 39, number 159 july 1982, pages 179-194 Evaluation of the Integral ftfu"e "\ u + x) ' du By K. S. Nagaraja and G. R. Verma Abstract. The representations of the above integral, in a power series form for small values of x and in an asymptotic form for large values of x, are given for integer values of n. In view of the usefulness of this integral, tabulated values are also presented for a wide range of values of x and p, and for a few values of n. 1. Introduction. Integrals involving exponential functions occur frequently in several problems in kinetic theory and generally in mathematical physics (cf. [l]-[5], [7]). An integral of the form /0 e~"(u + x)~x du has been discussed in [2], [3], [4], [7], and a particular integral of the form /<f e~u(u + x)~x du has been evaluated in [5]. This last integral is generalized and its representation, both in a power series form for small values of x and in an asymptotic form for large values of x, are given for integral values of n. In view of the usefulness of this generalized integral, tabulated values are also presented for a wide range of values of x and p, and for a few values of«. The integral (1) f (x,p)= fpu"e-"2(u + x)-ldu is considered, and its analytic representations are given for small as well as for large values of x. For small values of x, we can write fn{x, P)= rp f [u"~ - U"~2 + 2U"-3 J0 (2) + + (-\)rxru"-r-x + + (-\)"-x"x}e-"2du + (-\)"x"fpe-u\u + x)-]du. All the integrals except the last one in (2) can be expressed in terms of the incomplete gamma functions, and the last one can be evaluated as follows. If we write f0(x,p) = fpe-»2(u + x)-xdu and set y = f0 + e~x In x, then it can be seen that y satisfies the differential equation dv i e'pl 1 2 (3) -f + 2xy= /7rerfp + -(1 - e~x ), dx p + x x Received September 6, 1979; revised August 27, 1981. 1980 Mathematics Subject Classification. Primary 65A05, 65D20, 65D30. 179 1982 American Mathematical Society OO25-5718/82/O0OO-OO79/$O3.5O

180 K. S. NAGARAJA AND G. R. VERMA where Assuming a series solution of the form 2 fp _ 2 erf p = = J e " du. \JTT J0 00 (4) y = 2 akxk k = 0 for the differential equation (3), we obtain the coefficients ak. They are given below. 00 / l\n-l 2n (5) a0 = \np-^ 2 (~l) P v ' u 2, n n\ n= I (6) ax = 2ÍPe-u2du + e-p2/p, 0 (-1)' (7) 2k+2 k+\ 2(k+\)p2k+2 2(k+l)[(k+l)\]' k = 0,l,2,..., (8) ^+'=^TT+ (2k+l)p2k+x' k=^2,... The coefficients a2k+, and a2a: may also be expressed as follows: (9) a,hi^_(-i)v'2 (-ir+1ü)r fc=12 W ^+, (i)i ( )a 1 p2r+x, k 1,2,..., _(-!)*«(-l)v^y(-l/r! do) 2"" *' 2^! '= /,2r+, H)!V_L_ *=,. 2*! nr+l'».a---. r=0 where (a)k = T(k + a)/t(a). But for machine computations, it is more economical to use (7) and (8), rather than (9) and (10). Thus/n(x, p) is given by fr{x,p) = (Pune-"\u + x)'xdu Jo (11) =r(f 2 'p2) ~b(i ~hp2) +yai -hp2) +...+(-irft(f-f,=) + (-l)v[2a***-<r*2lnx],.n-\ +...+(-1rifY(i,^) where z^'s are given by (5), (6), (7), and (8); and y(a, x) is the incomplete gamma function given by (12) y(a,x)= fe-'ta-xdt. Jn

EVALUATION OF THE INTEGRAL f^u"e~u\u + x) ' du 181 2. An Alternative Approach. Since l/(w + x) = (1 x/(x + u))/u, it follows that (13) where f (x, p) = C _x(p) - xf _x(x, p), (14) Clearly, (15) C (p)=(pu"e-»2du. Jo fp 2 V77 C0{p)=J e~" du = erí(p), (16) Further, Cx(p) = 2(\-e-P2). (17) So (18) and (19) Cn+2{p) = -l2p"+[e-»2 + 2(n+\)C (p) ^2n+l x I n 1 -nip «nwe n~ ' 2k P k b = l0 (I), n-1,2k k=o(k+l)\ 1)n M)> In most cases it is easier to compute C by the recursion relation ( 17) than it is to use (18) and (19), and the same is true for/, using (13). Of course, if n is large, stability problems can arise. If the C are known, use of (13) in the forward (backward) directions is stable as n becomes large if x < 1 (x > 1). On the other hand, use of (17) is not stable in the forward direction as n becomes large. In (14), we use the exponential series to get (in hypergeometric notation) (20) Cn(p) n+l n+ 1 i*i / \ n + 1 2 H+ 1 + 1 which is convenient for p fixed and n large. Combining (13) and (17), we have (21) fn+fnï 2*/ + 2 2/n + 3 _ /T+'cn + 1 n + 1 n\c, pn+]e-pn+ 1 i^i 1 n + 1 + 1 This is advantageous since the need of C is bypassed. On the other hand, there are obvious stability and round-off difficulties. Still the expression is quite useful for moderate n and x provided sufficient round-off controls can be initiated. Repeated use of (13) gives (22) / = j,(-l)vc,.á.1 + (-l)v/0l k = 0 which is the same as Eq. (11) given above.

182 K. S. NAGARAJA AND G. R. VERMA (23) For large values of x and for p < x 1 00 i 00 fn{x, P) = ~ Zi x Cn + 2k 2 Li Cn + 2k+x, k = 0 k = 0 which is the same as Eq. (24) below. 3. Asymptotic Representation for Large x. When p < x, the function fn(x, p) can be written as (24) fn{x,p)=f fl" du= 2 ~rl ipu2k+ne-"2 du J0 x(l + u/x) k=0 x2*+1 'o - 2 ~Tk--iP«2k+n+le-uldu. k=0x2k+2jo Now we consider the odd and even values of n separately. Let n = 2m, h^p)=2-értjy 1 f..2k + 2 du 1 2 k = 0 x (Ik + 2m- \)(2k + 2m - 3) 3 lyv erf p 2* + m+l ~'x2k+1 (25) (2k + 2m)(2k + 2m - 2) 4 2 (1 - e''2) + e-pl t plk + lm plk + lm-\ e-"2 y k+y ' [ (2k + 2m)(2k + 2m - 2) (2k + 2m - 2r + 2) p 2k + 2m-2r + T Z 2i ) -,/ r2k + 2 If w = 2m + 1, (2k + 2m- \)(2k + 2m - 3) (2A: + 2m - 2r + 1) p^ + 2m~2r-\ 2r *" (26) 2 * = 0 L A=0 OO. 2- n/><2i+2'"+^-"2^ x'-^-'o (2Â: + 2m)(2A: + 2m - 2) 4 2 1 - <T (2A: + 2m + 1)(2A: + Im - 1) 3 Idv erf p 2* + m + 2 r2* + 2 e y. p2k + 2m+l p2k + 2m 2 " *~m (2fe + 2m + 1)(2A: + 2m - 1) (2k + 2m - 2r + 3) p 2 2'' k = 0 r=1 _2 oo k + m 1 2k + 2m-2r+\ 2k + 2 y y (2fe + 2m + l)(2fc + 2m - 1) (2k + 2m - 2r + 3) p^ + 2>n-2r+\ 2r v2* + 2

EVALUATION OF THE INTEGRAL fgune"(u + x)du 183 Asymptotic expansions for the case p > x have not been shown, although they can be easily derived. However, mention may be made about the difficulties associated with asymptotic expansions of functions of several independent variables such as p, x, and n that characterize the present problem. General methods for deriving the appropriate expansions are discussed in [6], [8], and [9]. The authors would like to express their appreciation for many illuminating comments and suggestions to the referee of the paper. 448 Merrick Drive Beaver Creek, Ohio 45385 Department of Mathematics University of Rhode Island Kingston, Rhode Island 02881 1. M. Abramowitz, "Evaluation of the integral /0 e~"2'x/u du," J. Math. Phys., v. 32, 1953, pp. 188-192. 2. A. Erdélyi, "Note on the paper 'On a definite integral' by R. H. Ritchie," MTAC, v. 4, 1950, p. 179. 3. E. T. Goodwin & J. Staton, "Tables of f? e~u\u + x)~' du," Quart J. Mech. Appl. Math., v. 1, 1948, pp. 319-322. 4. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962, p. 186. 5. K. S. Nagaraja, "Concerning the value of j{ e~" /(u + x) du," J. Math, and Phys., v. 44, 1965, pp. 182-188. 6. F. W. J. Olver, Asymptotic Expansions and Special Functions, Academic Press, New York, 1975. 7. R. H. Ritchie "On a definite integral," MTAC, v. 4, 1950, pp. 75-77. 8. N. M. Temme, "Uniform asymptotic expansions of confluent hypergeometric functions," J. lnst. Math. Appl., v. 22, 1978, pp. 215-223. 9. N. M. Temme, "The asymptotic expansions of the incomplete gamma functions," SI A M J. Math. Anal., v. 10, 1979, pp. 757-766.

K. S. NAGARAJA AND G. R. VERMA Û.50 2. 50 4. 50 0. 404 0.19t 0. 2 1. 182 1.390 0 1.554 1.579 1.589 1.593 1. 594 1. 594 1. 594 25003 47987 51896 53917 40054 67796 25510 35165 06624 32411 69318 78283 80858 81430 81239 81239 81430 81335 81239 81430 81335 0.22244239 7673512 982618 0.92654371 1892471 6599808 8777523 9686184 1.10026073 1. 10140324 1. 10174561 1.10183907 1. 10185432 1. 10185814 1. 10186005 1. 101859Ü9 1. 10185623 1. 10185814 1.10186005 1.10186195 1.10186291 0.15365773 0.34178519 0.56584454 0.70498991 0.78470379 0.82625818 0.84581006 0.85407192 0.85719633 0.85825127 0.85856795 0.85865402 0.85867333 0.85867757 0.85867e22 0. 85867858 0.85867852 0.85067682 0.85867882 0.85867886 0.85667780 0.8 0.11740285 0. 26674104 5221686 0.57163489 4174098 7893845 9667739 0.70425236 0.70714056 0.70812249 0. 70841950 0. 70849955 0.70851868 0. 70852298 0.70852375 0. 70852339 0.70852345 0. 70852345 0.70852351 0.70867401 0.70852345 0.09500092 0.21882963 0.37708789 8168701 0.54425645 0.57792646 0.59415781 0115069 0383701 0475606 0503554 0511231 0512835 0513109 0513276 0513288 0513312 0513276 0513365 0513377 0513377 0. 10 0. 50 2.2 5 4.2 5 1. 5 0.064 0. 151 0. 266 0.346 0.396 23 36 42 44 U. 445 45 45 45 45 45 45 45 45 45 45 45 32742 11786 93827 Í854S 10386 32035 71268 57915 86487 65600 89829 96553 98073 98460 98502 98514 98514 98496 98496 98484 98556 0.04862998 0.11544436 0.20679057 0.27136558 0.31206781 0.33490866 0.34630644 0.35135949 0.35334849 0.35404283 0.35425764 0. 35431713 0.35433161 0. 35433441 0.35433513 0.35433531 0.35433507 0.35433489 0.35433471 0.35433483 0. 35433549 0.032 0.078 0.142 0.189 0.219 0.236 0.245 0.249 1 1 1 1 1 1 1 1 1 1 1 1 1 68173 43828 65782 37039 53613 82004 60267 55833 13797 69617 87111 91957 93155 93423 93423 93465 93483 93447 93441 93393 93465 0.02461105 0.05940349 892212 0. 14551497 0. 16947830 0. 18338042 0. 19052434 0.19377470 0.19508415 0.19555110 0.19569808 0. 19573951 0.19574970 0. 19575191 0.19575226 0. 19575244 0. 19575238 0. 19575244 0.19575244 0. 19575197 0.19575208 Q.01973713 0.04780449 0.08809996 0.11817753 0.13805443 0.14968193 0.15570223 0.15846068 G.15957916 0.15998018 0.16010743 0.16014320 0.16015208 0.16015387 0.16015428 0.16015458 0.16015458 0.16015452 0.16015446 0.16015428 0.16015440 6.C 0.016 0.039 0.073 0.099 0.116 0.126 0.131 0.134 0. 135 06843 04446 0. 135 39600 0. 135 50818 0.135 53965 0.135 0. 135 0.135 0. 135 0.135 0.135 0.135 0.135 0. 135 47458 99564 96466 49666 47779 47051 67268 54758 54925 54949 54966 54966 54972 54955 54960 54960 7.Ü 0.01413765 0.03437996 0.06373960 0.08591968 0.1Ù074162 950249 0. 11408240 0. 11619973 0.11706561 0. 11737859 0.11747867 0.11750686 0.11751401 U. 11 751550 0.11751568 0.11751580 0.11751592 0.11751580 0.11751586 0.11751574 0.11751580 8.0 0.01238136 0.03014719 0.05599904 0.07560515 0.08875507 0.09655458 064507 0. 10254198 332000 360199 0. 103692 53 371792 372442 372591 372591 372605 372615 372615 372615 372621 372615 9.0 0.01101321 0.02684249 0.04593534 0.06750280 0.07931966 0.08634782 0.09004366 0.09176177 0.09246796 0.09272456 0.05280705 0.09283036 0.09283644 0.09283763 0.09283787 0.09283793 0.09283787 0.09283787 0.09283787 0.09283793 0.05283793 0.00991734 n.02419074 0.04505673 0.06096961 0.07169884 0.07809472 0.08146501 0.08303505 0.08368176 0.08391726 0.08399314 0.08401459 0.08401996 0.08402097 0.08402133 0.08402139 0.08402133 0.08402139 0.08402145 0.08402139 0.08402157

EVALUATION OF THE INTEGRAL l^u"e'u\u + x) ' du 185 0. 10 0.7 5 2. 50 3.no 4.2 5 4.7 5 0.2 0.018 0.085 0.224 0.3 52 46 6 0.540 0.556 0. 563 0.566 0. 566 0.567 0.567 0.567 0.567 0.56/ 0.567 0.567 0. 56/ 0.567 0.56/ 81764 29164 77663 13637 74289 96224 33589 54377 46309 06424 92636 17771 24411 25907 26176 26235 26255 26235 2o241 26205 26182 0.01069060 0.05419369 0. 15734977 962687 0.33925259 0.39150025 Ü.42107749 3566942 4157500 4436800 4516832 4540328 4546437 4547838 4548118 4548112 45481/2 454dl24 0. 44548112 4548088 4548190 0.00747251 0.03981648 0. 12177354 0.20725069 0.27600135 0.32214355 0. 34870112 0.36197056 0.36776292 0.36997920 0.37072438 0.37054486 0.37100202 0.37101565 0.37101841 0.37101885 0.37101907 0.37101863 0.37101901 0.37101889 0.37101855 0. 8 0.00574524 0.03149450 0.09950727 0.17293733 0.23343104 0.27474767 0. 29884565 0.31101239 0.31636798 0.31843185 0.31512953 0. 31933162 0.31939101 0.31940424 0.31940651 0.31940711 0.31940699 0. 31540693 0.31940734 0.31940681 0. 31940653 0.004666/4 0.02605814 0.08419275 0.14855766 0.20256764 0.23997229 0.26202553 0.27326298 0.27824306 0.28017426 0.28082507 0.28102368 0.28107780 0.28108566 0.28109199 0.28109235 0.28109229 0.28109205 0.28109294 0.28109181 C.28109211 O 4.2 5 1.5 0.00317645 0.01821086 0. 06') e 731 5 0. 110 06665 0.15266752 0. 18291724 0.20111829 0.21054476 0.21478248 0.21644557 0.21701616 0.21718717 0. 21/23413 0.21724430 0.21/24713 0.21724731 0. 217 24713 0.21724745 0.21724755 0.21724755 0.21724/49 0.00240765 0.01399892 0.04769936 0.08751369 0. 12268740 0.14308065 0.16357434 O. 171Ó9440 0. 1/538303 0. 17684317 0.17734855 0.17/50198 II. 1/754275 0.17755228 0.17/55413 0.17755467 1. 17/55467 0.17755479 0. 1/755473 0. 17755473 0.17755473 0.00162244 0.00557266 O.0333O655 13342 0.08821547 743765 0.11937540 0.12573751 0.12 666ee 0.12584075 0.13025177 0.13037763 0. 13041103 0.13041567 0.13042116 0.13042140 0.13042164 0.13042164 0.13042158 0. 13042158 0.13042156 4. 0 0.00122344 0.00727372 0.02559210 0.04818486 91060 0.08437556 0.05409028 0.05931523 174394 272563 307187 317660 320771 321468 321605 321617 321623 321635 321623 0. 10321635 321623 58196 0.00586521 0.02078055 0.03935676 551/1 0.06948811 0.07767606 0.08211040 0.08418500 0.08502847 0.08532768 0.08542049 0.08544576 0.08545172 0.08545297 0.08545321 0.08545321 0.08545321 0.08545315 0.08545327 0.08545327 2. 2 5 3. 50 U. i) Ü.000 82009 0.00491 3 13 Ü.01/49247 0.03326480 0.04795653 0.0590/555 0.066 1513 8 O.O70OO3O9 0.07181364 0.07255310 Û. 0 72 81643 0.072 89845 0.07292080 0.07292616 0.07292736 0.07292753 0.07252755 0.07252/53 0.0729274/ 0.07292753 0.07292753 70404 0.00422790 0.01510302 0.02880696 0.04163236 0.05138110 0.05761093 0.06101525 62141 0.06327963 0.06351477 0.06358826 0.06 360841 0.06361324 0.06361425 0.06361443 0.06361443 0.06361443 0.06361437 0.06361443 0.06361437 8.0 0.003 0.013 0.025 0.036 0.045 0.051 0.054 0.055 61676 71007 26804 40322 7e28C 46222 02665 07679 520C4 11311 32 557 39211 41037 41475 41565 415e5 41551 41552 41552 41552 41592 9.0 54873 0.00330525 0.01186254 0.02271910 0.03294581 0.04076734 0.04575468 0.04855751 0.04586782 0.05040750 0.05060127 0.05066206 0.05067877 0.05068280 0.05068365 0.050683 80 0.05068384 0.050683 86 0.05068386 0.05068384 0.05068388 49422 0.00298009 0.01071329 0.02054814 0.02983411 0.03695206 0.04153718 0.04406188 0.04526170 0.04575678 0.04593484 0.04599082 0.04600624 0.04600996 0.04601074 0.04601088 0.04601092 0.04601094 0.04601094 0.04601093 0.04601093

K. S. NAGARAJA AND G. R. VERMA 3. 25 4. 50 0.2 0.00121155 0.01321511 0.06564420 0. 14468104 0.22Ó71135 0. 29380119 0.33923274 0.36530548 0.37814857 0.3d362l69 0.38564845 0» 38630366 J.3J6<ie391 0.38653451 0.38654464 0. 38654649 0.386 54673 0.33654597 0. J8654691 0.38654697 J. 336 54661 69884 0.00861596 0.04765955 0.11125755 0. 18035877 0.23859382 0.27886897 0.30234617 0.31405151 0.31908/03 0.32056690 0.32157838 0.32175201 C.32179493 0.32180434 0.32180625 0.321b0643 0.32180643 0.3218Ufc55 0.32180661 0.32180673 45133 0.00640354 0.03753531 0.09075773 0.15045523 0.20150752 0.23807865 943154 0.27018396 0.27484651 0.27655941 0.27717263 0.27733554 0.27737713 0.27738565 0.27738744 0.27738786 0.27733804 0.27738786 0.27738798 0.27738752 0.8 37889 0.00509784 0.03099367 0.07675856 0.12931526 0.17539585 0.20822304 0.22780383 0.23774701 0.24208903 0.24373031 0.24426967 0.24442416 0. 24446315 0.24447137 0. 24447310 0.24447334 0.24447322 0.24447340 0.24447322 0.24447352 30834 0.00423525 0.02640670 0.06655061 0.11349213 0.15522027 0.18527013 0.20335078 0.21259785 0.21666014 0.21820354 0.21871281 0.21885955 0.21889639 0.21890450 0.21890599 0.21890610 0.21890628 0.21890622 0.21890634 0.21890640 C.1Q 2. 75 4. 75 O.'IÜO 0.002 U.019 1. 5 0.0 50 0.037 0. 120 0.145 0. 160 0. 168 0.172 0.173 0. 173 0. 174 0. 174 0.174 0. 174 0.174 0.174 0.174 0. 1/4 0. 174 21041 97715 28976 00827 05e54 81 778 62213 75682 66714 16605 50506 95633 08597 11888 12603 12746 12770 12770 12788 12782 12786 159 70 0.00229561 0.01520072 0.04008095 0.07068509 Ü.O9903234 0.120151T4 0.13322502 0.14007533 0. 14314812 0.14433682 Ü. 14473540 0.14485180 0. 14488143 0. 14488804 0.14488918 0.14488947 0.14468953 0. 14488559 0.1448894/ 0.14488959 0.001 0.010 0.028 0.051 0.072 0.089 C.099 4 10776 57482 67846 70752 41361 87957 16116 40034 84062 0. 107 31184 8 27845 0. 10 e60598 8 70242 0. 108 72716 C 7327C 8 73271 8 733P3 0. 108 73385 8 73401 0. 108 73401 0. 106 73395 08131 0.00119855 0.00e23114 0.02236875 0.04041724 0.05769125 0.07093841 0.07935327 0.08386523 0.08593154 0.08674604 0.08702391 0.08710629 0.08712751 0.08713222 0.08713329 0.08713235 0.08713347 0.08713341 0.08713347 0.08713347 06529 96742 0.00669677 0.01832464 0.03 330154 0.04775330 0.05891573 0.06606203 0.06991625 0.07169181 0.07239556 0.07263690 0.07270879 0.07272738 0.07273149 0.07273245 0.07273245 0.07273251 0.07273257 0.07273251 0.07273257 0. 10 0. 75 3. 50 6.0 0.005 0.015 0.028 0.040 0.050 0.0 56 0.059 0.061 05454 81103 6446Ö 51963 31845 74075 39141 59563 95946 51553 13548 34874 41249 42509 43288 43361 43372 43372 43370 43374 43377 04683 69dl7 0.00487834 0.01345968 0.02463354 0.03552624 0.04402349 0.04950739 0.05249156 0.05387712 0.05443044 0.05462147 0.05467878 0.05469373 0.05469713 0.05465775 0.05469789 0.05469790 0.05469788 0.05469793 0.05469794 8.0 04103 61288 0.00429523 0.01188266 0.02179771 0.03149621 0.03908632 0.04399976 0.04668131 0.04792980 0.04842966 0.04860265 0.04865465 0.04866825 0.04867139 0.04867201 0.04867211 0.04867212 0.04867211 0.04867209 0.04867214 9.0 Q.00003651 54616 0.003e3666 0.01063651 0.01954768 0.02828798 0.03514597 0.03559647 0.04203111 0.04316718 0.04362302 0.04378113 0.04382876 0.04384123 0.04384407 0.04384465 0.04384474 0.04384475 0.04384473 0.04384469 0.04384476 03285 49255 0.00346656 0.00962697 0.01771887 0.02567327 0.03192795 0.03599527 0.03822461 0.03526687 0.03968578 0.03983136 0.03987527 0.03988678 0.03988544 0.03988996 0.03989003 0.03989004 0.03989005 0.03985002 0.03989005

EVALUATION OF THE INTEGRAL f u"e " ( U + ) du 187 2. 75 0.2 0.002 0.022 0.072 0. 144 0.219 0. 281 0.323 0.347 0.3 58 0.363 0.365 0. 365 0. 365 0.365 0.365 0.365 0.365 0.365 0. 365 0. 365 08903 37428 81143 51757 12981 18035 19743 22109 05400 61939 28806 09275 61/22 76134 79472 80151 80294 80324 60318 80330 80205 05180 0.00157092 0.01687645 95090 0.11732858 0.18250448 0.23745679 0.27534401 0.29710376 0.30770892 0.3121327e 0.31372249 0.31421727 0.31435050 0.31438220 0.31438857 0.31438977 0. 31438845 0.31438954 0.31439024 0.314390ie 03654 0.00117518 0.0134150e 0.04699928 0.09919667 0. 15675747 0.20619706 0.24062377 0.26061457 0.27043563 0.27455980 0.27605035 0.27651644 0.27664274 0.27667254 0.27667868 0.27668017 0.27668005 0.27667987 0.27668017 0.27667553 0. 8 02823 93903 0.01114537 0.04004719 0. 00601593 0. 13762534 0.18246603 0.21403962 0.23252732 0.24167275 0.24553531 0.24693835 0.24737883 0.24745869 0.24752700 0.24753261 0.24/53422 0.24753392 0.24753404 0.24753410 0.24753404 02300 78202 0.00953358 0.03490334 0.07597989 0.12272161 0.16377413 0.19293189 0.21012682 0.21868342 0.22231575 0.22364066 0.22405833 0.22417235 0.22419553 0.22420496 0.22420639 0.22420627 C.22420609 0.22420633 0.22420627 0. 10 0. 75 2. 50 3.5'J 4.OU 4.2 5 4. 7 5 1. 5 015 72 0.00,155159 0.00700563 0.02644150 0.05888421 0.09671515 U. 13061094 0.15503729 0.16972365 0.17709464 0.18025577 J.18141925 0.181 78350 0. 18139037 0. 18151475 0. 18191981 0. 13192071 0. 18192059 0. 181920 7 7 0.18192083 0.18192083 01194 42605 0.00 5 53 8 84 0.02129211 0.04810187 0.07587714 0. 1U374212 0. 12983257 0. 14257365 0. 1490473 7 0. 15184563 0.15288264 0.15321404 0.15330571 0. 15332776 0.15333241 0. 15333319 0.15323325 0.15333325 0. 15331331 0. 15333337 8:3888«S8* 0.00390492 0.01533018 0.03523127 0.05530205 0.08156091 0.09808155 0. lce20186 0. 11340833 0. 11568397 0.11653556 0.11681062 0.11688751 0.11650587 0.11690980 0.11691040 0.1169105e 0.11691052 0.11651058 0.11691056 00609 22310 0.00301565 0.0119/895 0.02780307 0.04717706 0.065?90dl 0.07ee6564 0.08726358 0.051617(4 0.05353524 0.09425807 0.05445279 0.09455872 0.094574Í3 0.09457809 0.09457874 0.09457874 0.05457868 0.09457680 0.0945/e74 00489 18019 0.00245642 0.00983077 0.02256451 0.03917562 0.05444588 0.06597233 0.07314301 0.07688463 0.07854158 0.07916533 0.07937407 0.07943189 0.07944596 0.07544894 0.0/544942 0.07944547 0.07944542 0.07944953 0.07944559 o.oco 0. coo 0.002 0.008 Ü.0Í9 0.033 0.046 0.066 0.067 0. C68 00408 1511? 07216 23616 56151 45756 69604 70914 56778 24800 70664 26144 44312 49450 50/13 50570 51017 51023 51017 51017 51025 00351 13013 o.ooi75iea 0.00723616 0.01/03738 0.02925841 0.04088014 0.049/3115 0.05528364 0.05820379 0.05950652 0.06000355 0.06016674 0.06021306 0.06022449 0.06022684 0.06022731 0.06022737 0.06022735 0.06022735 0.06022741 8.0 00307 11426 0.00157839 0.0063926e 0.01505047 0.02557234 0.03635385 0.04428457 0.04527404 0.05190532 0.05306225 0.05353241 0.05360057 0.05372272 0.05373313 0.05373529 0.05373571 0.05373575 0.05373575 0.05313576 0.05373575 00274 10184 0.00141036 0.00572535 0.01354302 0.02335026 0.03273080 0.03S91448 0.04444463 0.04663506 0.04791236 0.04832372 0.04845538 0.04 849803 0.04850755 0.04850559 0.04850598 0.04851004 0.04851002 0.04851002 0.04851006 00246 05185 0.00127467 0.00518420 0.01228351 0.02120928 0.02976489 0.03633017 0.04047846 0.04267515 0.04366158 0.04404030 0.04416542 0.04420112 0.04420991 0.04421182 0.04421218 0.04421221 0.04421221 0.04421221 0.04421223

K. S. NAGARAJA AND G. R. VERMA 1. 25 1. 50 2.no 2. 75 5.no O. 2 o. coo 0.00 0.040 o. i :.z 0. 167 0.2 tz 0.340 0.2 84 U.4C9 20 24 0.»?6 26 U.t26 26 26 O.A2o?6 26 26 00703 46155 68721 35361 2"455 55924 48/22 35426 79167 08861 32385 75700 L'jlll 65827 31056 83566 8378o 83852 83840 83852 83853 00411 30844 0.00649892 0.03207676 0.08 518910 o.15843356 0.23372/16 0.29486030 0.33337173 0.357/2890 0.36814651 0.37228590 U.37369621 0.37410933 0.37421602 0.37423992 0.37424529 0.3/424296 0. 37424296 0.37424284 0.37424326 00251 23170 0.00515606 0.02665756 0.07260239 0. 13735682 0.20500761 0.26062232 0.29784405 0.31855106 0.32826575 0.33214505 0.3334725? 0. 33386397 0.33396359 0.33398771 0.33399266 0.33399105 0.32359055 0.33359055 0.33359141 0. 3 00225 18558 0.00433321 0.02281941 0.06330442 0. 12133491 0.18276685 0.23376495 0.26819026 0.28747410 0.29657322 0.30022466 0.30145323 0. 30185062 0.30154598 0. 30196708 0.30197269 0. 30197167 0.30157155 0.30197155 0.30157157 00183 15479 0.00371592 0.01995380 14055 371339 0.16495168 0.21206450 0.24408484 0.26212865 0.27068532 0.27413523 0.27532399 0.27567714 0.27576774 0.27578813 0.27579314 0.27579254 0.27579236 0.27579308 0.27579254 0.2 5 2.7 5 0.002 O.U15 0.043 0.086 0.132 1. 5 0. 172 0.199 0. 215 0.222 0. 225 0.226 0.227 0.227 0. 2 27 0.227 0.227 0.227 0.227 0.22/ 00125 10943 74106 19488 7941 7 36224 80928 36555 62472 17164 61765 64590 69905 01389 09515 11243 11712 11867 11736 11754 11 778 00055 08463 0.00217181 0.01227292 0.03591667 0.07168076 0.11124164 0. 14533126 0.16906291 0. 18271810 0. 18931001 0. 19 20092 7 0.192553ee 0. 19323784 0. 19321175 0. 19332826 0. 19333160 U.19333255 0.19333291 0. 19333190 0. 19333220 00064 05824 0.00153472 0.00686660 0.02642665 0.05352663 0.08404332 0.11075163 0.12960452 0.14058781 0. 14594954 0. 14ei6695 0. 14855004 0.14918751 0.14524561 0. 14926362 0.14926642 0.14926650 0.14926744 0.14926666 0. 14926708 O.0QQQO049 04440 0.00118672 0.00694131 0.02090822 0.04272683 0.06756288 0.08951795 0. 10515606 0.11434203 0. 11886060 0. 12074208 0.12141097 0.12161499 0.12166870 0.12168080 0.12168318 0. 12168372 0.12168407 0. 12168372 0. 12168413 00039 03587 56739 0.005/0326 0.01729750 0.03555714 45678 0.07513499 0.08849519 38935 029352 C.1C152B09 251164 269040 273772 274339 C.10275054 275078 275120 275090 275102 1. 50 6. C n.ooo o.uoo o.uoo 0.004 0.014 0.030 0.048 0.064 0.076 0.083 0.086 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.0 88 0.0C8 00033 03009 81650 84011 75141 44982 54954 74203 40368 32445 76165 20617 72384 88298 92518 93478 93675 53710 53728 53698 53716 00028 02 552 70633 0.00420394 0.01285875 0.02662634 0.04256514 87869 0.06722492 0.07238615 0.07645607 0.07775021 0.07821536 0.07835871 0.07839680 0.07840556 0.07840725 0.07840765 0.07840776 0.07840747 0.07040765 00025 02276 62236 0.00371560 0.01139667 0.02365640 0.03765501 0.05072064 0.06001819 0.06557012 34356 0.06951565 0.06593800 0.07006842 0.07010317 0.07011122 0.07011282 0.07011300 0.07011318 0.07011306 0.07011306 9.0 8:8888i ll 55623 0.00332692 0.01023322 0.02128278 0.03414897 0.04576690 0.05420676 0.05926104 0.06179039 86144 0.06324810 0.06336778 0.06339967 0.06340712 0.06340855 0.06340885 0.06340891 0. 06340891 0.06340879 00020 01830 50281 0.00301515 0.00928535 0.01934221 0.03107730 0.04169540 0.04942582 0.05406097 38564 0.05737175 0.05772635 0.05783889 0.05786e4l 0.05787529 0.05787664 0.05787689 0.05787701 0.05787696 0.05787691

EVALUATION OF THE INTEGRAL ffi Une " (u + x) du 189 0. 10 1. 2 5 3. 50 4. 51) 0. 2 00058 09441 0.00349796 0.02392180 Ü.07960975 0. 17469484 0.29120827 0102309 83e6610 0.53514944 0.56163257 0.57317674 0.5/745945 0.57881953 0.579 19151 0.57927853 0.57929790 0.57930088 0.57930166 0.57930076 0. 579 3U118 00034 06342 O.O0263584 0.01516183 0.06619304 0. 14864126 221467 0.35114598 2667675 7387451 9343536 921327 0.51323271 0.51451540 0.51486766 0.51495141 0.51496840 0.5 1497122 0.51497197 0.51497138 0.51457197 00024 04778 0.00211651 0.01599800 70725 0.1298005t 0.2227OO82 0.31272016 0.38211763 2583543 4873804 5884057 6262753 6384072 6417511 6425480 6426552 6427395 6427453 6427375 6427423 0.8 00019 03633 o.ooi76ee4 0.01373701 0.04962515 0.11514676 0.19950254 0.28208160 0.34627211 0. 38698673 0843892 1794842 2152756 2267859 2255670 2307270 2308724 0. 42305129 2309177 0. 42305093 2309147 00015 03201 0.00151949 0.01203872 0.04412822 350096 0.18075389 702858 0.31674039 0.35483706 0.37501174 0.38399357 0.38738716 0.38848168 0.38878518 0.39885796 0.3e887310 0.38887584 0.38887626 0.38887560 0.36887602 î. 50 5.Ou 0.C01 0.0U9 0.034 0.082 1. 5 0. 146 0.210 0. 261 0.294 0.311 0.319 0.322 0.323 3.3?3 0.323 0.323 0.323 0. 323 0.323 0.323 00010 02266 12383 20020 57750 67105 4915S 544ee 38/17 21043 770 74 65859 66229 63784 91030 97574 58975 59195 99237 9917e 55166 0. 00000008 01754 89178 0.00744667 0.02843542 o.06se53oe 0.12322223 0.17843097 0.22269857 153130 0.26707733 0.27410507 0.27680296 0. 27768296 0.27792990 0.27798921 0.27800220 0.27800304 U.27800471 0.27800399 0.27800405 00005 01208 63122 0.0053927C 0.02098882 0.05162828 0.09357540 0.13683826 0.17201060 0.19520414 0.20784944 0.21362680 0.21586043 0.21659642 0.21680427 0.216e5465 0.21666566 0.21686721 0.21te6765 0.21666745 0.21686J27 00004 00922 n.00048852 0.00422729 0.01663584 0.04130744 0.07545412 0. 11102176 0.14015566 0.15555930 0. 17025614 0.17515898 0.17706674 0.17765e51 0. 17787350 0.17792249 0.17793190 0.17793345 0.17793369 0.17793363 0.17793328 O.OO0UO0O3 00/45 39845 0.00347622 0.013 77522 0.03442501 0.06322169 0.09341854 0.11334 794 0.13502038 0.14422541 0.14848512 0.15015256 0.15070665 0.15086478 0.15090352 0.15091137 0.15091324 0.15091342 0.15091336 0.1509131S. 10.25.50.75.00.25.50.75.00.25.50.75.00.25.50.75.00. 25.50.75.00 6.0 0.00(1 0.002 0.011 0.0 29 0.054 0.080 2 0. 11/ 0. 125 0.128 0.13U 0.130 0.131 0. 131 0. 131 0.131 0.131 0.131 0.131 0000 3 00625 33643 95182 76023 51577 '.0603 64133 40233 02013 12745 88885 36776 86098 00213 03706 04439 04576 045ea 04580 04564 00002 00 53 8 29111 0.00256493 0.01025746 0.U2583O37 0.04774984 0.07094258 0.09024972 326368 0.11050498 0.11387491 0.11520356 0.11564803 0.11577541 0.11580694 0. 11581367 0.11581492 0.11581510 0.11581498 0.11581492 8.0 Q.OQ0Q00O2 0.QQQ00002 00473 00421 25656 22933 0.00226771 0.00203223 0.00505532 0.00816577 0.0229635C 0.02066963 0.04254574 0.03836508 0.06332844 0.05719146 0.08067506 0.07294554 0.09240627 0.08361763 0.09854866 0.08958405 200095 0.09237349 320705 0.09347705 361165 0.09384894 372776 0.09395576 375643 0.09398216 376265 0.05358782 376376 0.09398878 37640C 0.C5398856 376394 0.09398896 376362 0.09398878 0.OQ0QO002 00380 20733 0.00184105 0.00741521 0.01879254 0.03493284 0.05213951 C.06656629 0.07635736 0.08184105 0.08440930 0.08542764 0.08577043 0.08586525 0.08589381 8589500 0.08589989 0.08590007 0.08590001 0.08589589

K. S. NAGARAJA AND G. R. VERMA 0.7 5 0. 2 O.0O000005 01955 0.00146190 0.01478853 0.06437927 0. 17205834 0.33241856 0.51046044 6512263 0.77348346 0.83597219 0.86605680 0.87827456 0.88249296 0.88373792 0.88405347 0.88412237 0.88413602 0.88413811 0.88413805 0.88413638 00003 01346 0.00110716 0.01190816 0.05382412 0. 14746082 0.28977442 5020503 0.59122515 9096303 0.74892330 0.77700663 0.7884/808 0.79245043 0.79362905 0. 79392898 0.79399514 0.79400736 0. 79400927 0.79400724 0. 79400820 00002 01016 89155 0.00597410 0.04627702 0. 12911665 703961 0303284 0.53262430 2501192 7905575 0.70538729 0.71618586 0.71595223 0.72107095 0.72135663 0.72141981 0.72143175 0.72143370 0.72143406 0.72143245 0.8 00002 00616 74642 0.00858329 0.04060124 0.11487997 0.23105770 0.36499965 0487729 0.57092375 2154734 4633310 5654439 6011399 6117665 6145122 6151184 6152316 6152513 6152495 6152400 00001 00682 64201 0.00753416 0.03617315 349602 0.20990610 0.33363593 4515526 0.52567565 0.57328647 0.59669765 0637515 0977455 1079001 1105114 1110502 1112064 1112139 1112189 1112076 8:iS 1.5 Ü. 000 00001 00483 47575 0.00577258 0.02843509 0.08299025 0. 17092258 0.27484733 0.36931437 3915706 8064190 120014 576145 0.51279896 0.513/1002 0.51394528 0.51359779 0.51400810 0.51400948 0.51400971 0.51400870 g.googooqi 00374 37794 0.00467953 0.02343050 0.06929106 0.14421576 0.23380260 0.31649846 0.37745023 1414350 3247151 4016033 4288975 4371575 0. 44392557 4397801 4358743 4398880 4398874 4398880 00000 00258 26784 0.00339475 0.01733488 0.05211242 993356 0. 18014944 0.24586365 0.29490000 0.32474565 0.339ei019 0.346ie461 0.3484Í675 0. 3491625e 0.34934443 0.34938550 0.34939313 0. 34939474 0.3493946e 0.34939486 g.ooogooog 00197 20742 0.00266370 0.01375794 0.04176748 0.08884370 0.14657730 0.20109701 0.24211562 0.26727319 0.28005493 0.28549629 0.28745985 0.28806096 0.28821874 0.28825444 0.28826141 0.28826267 0. 28826249 0.20826290 00000 00159 16525 0.00219179 0.01140525 0.03485227 0.07455152 0.12357134 0.17015538 0.20541012 0.22715062 0.23825246 0.24300152 0.24472219 0.24525130 Ü.24539065 0.24542242 0.24542868 0.24542964 0.24542981 0.24542559 6.0 o.io o.ogg 0.001 0.009 0.029 0.064 0. 106 0.147 2.2 5 0. 178 2. 50 0.197 0.20 7 0.211 3. 25 0.213 0.213 0.213 0.213 0.213 0.213 0.213 0.213 00000 00134 14294 86195 73554 90271 22400 81570 48162 39211 5325S 34507 55816 08929 56155 68688 71521 72086 72175 72199 72187 o.ooooogoo 00115 12371 0.00161841 0.00849909 0.02618469 41136 0.09406549 0.13014728 0.15766692 0.17476314 0. 18355459 0.18734127 0.18871965 0. 18914670 0.18925971 0.18928564 0.18929052 0.18929130 0.18929154 0.18929148 8.0 00000 00101 10905 0.00143122 0.0O753e73 0.02328927 0.05029425 0.08403677 0.11646307 0.14126283 0.15670925 0.16467196 0.16810971 0.16536374 0. 16975325 0.16985643 0.16988005 0.16988474 0.16588552 0.16988565 0.16588581 9.0 00000 00090 09749 0.00128284 0.00677341 0.02097060 0.04537451 0.07594150 538524 0. 12795419 0. 14204121 0. 14931 786 0.15246546 0. 15361619 0.15397388 0. 15406907 0.15405088 0.15409459 0. 15409577 0.15409607 0.15405625 00000 00031 08315 0.00116234 0.00614517 0.01907191 0.04133179 0.06926972 0.09623313 0.11693937 0.12988704 0.13658661 0.13948941 0.14055216 0.14088309 0.14097130 0.14059157 0.14059550 0.14099604 0.14099634 0.14099652

EVALUATION OF THE INTEGRAL fi U "e ' " ( U + ) du 191 4. 25 0.2 0.009 0.053 J. 1/6 0.397 36 0.975 1.205 1.352 1.4 31 1.466 1.479 1.483 1.464 1.484 1.484 1.484 1.484 1.484 00000 00432 62 73 9 41758 85637 28360 59314 81711 53/02 21641 79274 25057 12835 20990 38032 50375 78127 83276 84135 84325 84802 00000 00292 47690 0.00761203 0.04520261 0.15171111 0.34816706 0682682 0.87252140 8352852 1.22041798 1.29365826 1.32639406 1.33872 795 1.34267616 1.34375477 1. 34400749 1. 34405804 1.34406567 1.34406662 1.34407043 00000 00221 38484 0.00639083 0.03696606 0.13322526 0.30965276 0.54708970 0.78943676 0.98490703 1.11255455 1.18122959 1.21206856 1.22373676 1.22748566 1.22651161 1.22875404 1.22880268 1.22880536 1.22881031 1.22881603 0.8 00000 00178 32263 0.00550866 0.03425127 0. 11879152 0.27923030 9690968 0.72110587 0.90317512 2274990 8740711 1.11654758 1. 12761 784 1. 13118458 1. 13216496 1.13235670 1. 13244343 1.13245010 1.13245010 1.13245583 00000 00149 27776 0.00484112 0.03055509 719925 417006 5527303 6385591 0.83423799 0.94670117 776291 3540325 4593372 4933643 5027390 5049610 5054092 5054760 5054855 5055332 1. 50 I. 5 00000 00105 20614 0.00371641 0.02407959 0.08620965 0.20769250 0. 37063805 0.55426863 0.70111769 0.79902381 0.85265619 0.87712336 0.336 50978 0.88956165 0.890 40/20 0.89060875 0.89064968 0.89065685 0. 89065778 0.89065820 00000 00082 16389 0.00 301621 0.01987123 0.07211322 0.17564517 0.32130295 7601376 0 501271 9169405 0.73951381 0.76146203 0.76992828 0.77269495 0.77346486 0.77364516 0. 77368689 0.77369366 0.77369469 0. 1 Z3Ó9453 o.ocgooooo 00056 11626 0.00219103 0.01472757 0.05435796 0.13427585 0.2484554Í 0.37141937 7521240 0.54573667 0.58502805 0322827 1030726 1263317 1329144 1344892 1348110 1348712 1348784 1348832 o.ooooogoo 00043 09008 0.00172048 0.01170046 0.04362511 870200 0.20255953 0.30462217 0.39145052 5085275 8424041 5978572 586849 0.5C788242 844949 858670 861508 862032 662080 862110 00000 00035 07352 0.00141634 0.00970572 0.03643404 0.09131514 0.17105156 823063 0.33286172 0.38423210 1319788 2676473 3209678 3386530 3437070 3449223 3451750 3452205 3452293 3452275 i:ü2 5 fi? 6.0 o.ooo o.ono n.ooi 0.008 0.031 0.078 0. 148 U.224 0.289 0.3 34 0.3611 0.3 72 0.377 I). 3 78 0.379 0.379 0.3 79 0.379 0.3 79 0.379 00000 00029 06211 20360 29257 27913 73291 01395 12074 55919 78677 38971 42496 17098 75408 20320 31257 33534 33946 34017 34017 00000 00025 O.OOO05376 0.00104642 0.00723857 0.02740255 0.06919/42 0.13044983 0. 19797981 624233 0.29664010 0.31957906 0.33029345 0.33467001 0.33609974 0.336 5045e 0. 33660567 0.33662641 0.33663034 0.33663066 0.33663076 8.0 00000 00022 04735 92556 0.00642233 0.02438114 0.06172306 0.11661446 0.17730510 0.22980576 0.26631027 0. 28708720 0.29690516 0.300796ei 0.30210072 0.30247225 0.30256313 0.30258220 0.30256554 0.30258632 0.30258644 9.0 o.oooogogo 00020 04237 82973 0.00577154 0.02195999 0.05570652 543394 0. 16054314 0. 20832574 0.24161440 0.26060212 0.26959223 0.27316219 0.27436060 0.27470243 0.27478629 0. 274803 94 0.27480704 0.21480757 0. 27480763 00000 00018 03832 75188 0.00524051 0.01997633 0.05075901 0.09621048 0.14667851 0.19051838 0.2211148/ 0.23859680 0.24688774 018519 129384 161105 168836 170475 170773 170827 170627

K. S. NAGARAJA AND G. R. VERMA 0.5U i. 00 4. OU 0.2 00000 00095 2/463 0.00613066 0.04615315 0. 18593347 9218148 0.96253914 436555 1.98986626 2.33868885 2.54346466 2.64309120 2.68368145 2.69765377 2.701 73645 2.70275457 2.70297718 2.70301723 2.70302391 2.7030267 7 00000 00065 2093 5 0.0049693 7 0.03808338 0.160 50 5 83 3243325 0.85677195 1.35057640 1.79749966 2.12108040 2.31225105 2.40575790 2.44403U76 2.45726013 2.46113682 2.46210957 2.46231842 2.46236038 2.46236324 2.46236801 00000 00045 16921 0.00417968 0.03358463 0.14125508 0.38578038 0.77204825 1.2259225e 1.63596601 1.94171429 2.12097740 2.20907307 2.24528313 783920 2.26153374 2.26245975 2.26266003 2.26265913 2.26270350 2.26270560 0.8 00000 00039 14200 0.00360725 0.02956345 0. 12615705 0.34831572 0.70277202 1.12269552 836945 104710 1. 95975977 4307365 7743454 8938313 9290791 5379387 539e651 5402466 9402752 9403038 00000 00033 12235 0.00317305 0.02640533 0.11399096 0.31752556 n.64502317 3572750 1.35667034 1.66254711 1.82195091 1.90090847 1.93359375 1.54459302 1.94836521 1.94921303 1.94939804 1.54943523 1.54544000 1.94944191 1.5 O.OOOÇOOOfi 00023 0909Ü 0.00243956 0.02084504 0.09187567 0. 26016 110 0.53534466 U. du81 5369 1.1792326U 1.41071415 1.550729/5 1.62063122 1.64976311 1.65958745 1.66303062 1.663e0024 1.66396809 1.66399956 1.66400337 1.66400715 00000 00018 07232 0.00198174 0.01722197 0.07696360 0.22040945 5769477 0.74755841 20e8547 1.22535201 1.35068798 1.41339207 1.43967056 1.44893932 1.45171070 1.45241737 1.45256901 1.45259952 1.45260429 1.45260334 5.U o.ooooogoc 00012 05134 0.0014410e 0.01278165 0.05811632 0.16887154 0.35452015 0.58532935 0.80527204 0.97204173.07463074.12663174. 14860435. 15641403.15876464.15936852.15949917. 15552356.15952969.15952872 00000 00010 03579 0.00113226 0.01016260 0.04668984 0.13689172 0.28990346 6105961 6510701 0.80568057 0.89275944 0.93717265 0.95605260 0.96280009 0.56484202 0.96536869 0.96548223 0.96550453 0. 96550977 0.56551073 00000 00008 03249 93245 0.00843459 0.03902020 0.11510348 0.24504340 0843362 0.56660044 8809104 0.76373023 0.80249196 0.81904268 0.82498181 0.82678646 0.82725376 0.82735491 0.82737440 0.82737941 0.82738024 10 25 50 75 00 25 50 75 00 25 50 75 00 25 50 75 00 25 50 75 00 6.0 o.oqoggggo O.OOOOOOOt 02 74 5 79260 0.00720895 0.03351561 0.09930241 0.21221703 0.35486265 9355352 0052311 6737938 0.70176572 0.71649873 0.72130313 0.72341955 0.72393916 0.72393209 0.72394812 0.72395271 0.72395331 00000 00006 02376 68923 0.00629437 0.02937248 0.08731771 0. 18715191 0.31372 744 3721300 0.53276247 0.59266347 2356216 3683730 4162852 4309275 4347410 4355856 4357316 4357704 4257765 8.0 o.gooogooo 00005 02055 60971 O.OOÍ58575 0.02614115 0.07751515 0.16738498 0.28114474 0.39243007 7676275 0.53301972 0.561072J7 0.57315207 0.57752198 0.57885935 0.57920643 0.57928592 0.57529545 0.57930315 0.57530368 9.0 o.gooooooo 00004 01873 54664 0.00502055 0.02355050 0.07034129 0. 15139627 469714 0.35597616 3471372 8429811 59e586 0.52106750 0.52508324 0.52631521 0.52663672 0.52670819 0.52672076 0.52672434 0.52672458 00000 C.00000004 01694 49540 0.00455523 0.02142708 0.06410568 0.13819695 0.23279529 0.32572359 0.39809442 4374728 6743774 7767353 ei38893 8253018 8282784 8289496 8290652 8290992 C.48290598

EVALUATION OF THE INTEGRAL /<f W "e ~ " ( M + ) du 193 N = fc» 0.2 00000 00021 12207 0.00405902 0.04038458 0. 20045524 2541103 1.39313221 2.40846062 3.437 79564 4.263U7392 4.79794121 8266926 5. 208 69 541 5.25551937 5.27021503 5.27414854 5.2/503872 5.27521324 5.27522659 5.27524471 00000 00014 09326 0.00329740 0.03406935 0.17347580 0.55087423 1.24301147 2. 16910648 3. 11676975 3.88238045 4.38171864 4.64898491 4.76782608 4.81214523 4. 826 10893 4.829e5401 4.83070660 4.83067158 4.83088493 4.83090401 O.OOCO0O00 00011 g7547 0.00277735 0.02547232 0.15292901 5237460 1.1224908e 1.97378254 2.85180052 3.56578541 3403167 4.28584385 4.39826584 4.44034672 4.45364475 4.45722103 4.45803738 4.45815565 4.45820904 4.45822611 0.8 00000 00009 06335 0.00239935 0.0259724/ 0.13675648 4515502 2350044 1.81117821 2.62908554 3.29797459 3.73878193 3.57682858 8345228 4.12354565 4.13623810 4.13965607 4. 14044189 4. 14055639 4.14060879 4. 14062Î00 00000 00007 05465 0.00211210 0.02321789 0.12369102 0631759 0.94069105 1.67360592 2.43910885 6826496 3.48466492 3.71037674 3.81184964 3.85005665 3.86219692 3.3654/470 3.86622506 3.86637688 3.86639118 3. 96640739 8-.251 o:? 5 1*25 1 50 I:ü2 5 btt m 1.5 C.000 0.001 0.018 0.099 0.333 0.782 1.407 66 2.614 2.980 3. 180 3.270 3.305 3.316 3.318 3.319 3.319 3. 319 3.319 ooçpo 0000 5 04064 62561 35563 86836 60553 70274 10545 52886 74037 52406 3 4267 79773 06992 01715 9935/ 67545 80991 83757 6394d 8:88888882 O.OOC03235 0.00132168 0.01517928 0.0837546C 0.28302830 7032957 1.21421814 1.79401016 2.27909470 2.60524654 2.78450298 2.86610222 2.89717102 2.90714264 2.90986252 2.91049250 2.91061592 2.91063690 2.91062832 o.oogogggo 00003 02298 96151 0.01127811 0.06333286 0.21723135 0.52055856 0.55335050 1.41996002 1.81467523 8272839 2.23139477 2.29962826 2.32580757 2.33427143 2.33656886 2.33712959 2.33723736 2.33725548 2.33724685 g.gggooogo 00002 01782 75613 0.00897282 0.05092285 0. 17628026 2610532 0.78494287 1. 17534828 808239 1.73560619 1.66255937 1.92123127 1.94384956 1. 95119953 1.95322323 1.95365911 1.95379257 1.95380688 1.95380116 8:88888882 01455 62289 0.00745022 0.04258121 0.14832878 0.36050183 6717374 277328 1.29035091 1.48799133 1.59883022 1.65023041 1.67013531 1.67663193 1.67842960 1.67885399 1.67893696 1.67895126 1.67894459 0. 50 1. 25 2. 50 6.0 0.006 0.036 0.128 0. 312 0. 580 0.874 1.127 1.302 1.400 1.446 1.464 1.470 1.471 1.472 1.472 1.472 1.472 00000 0O001 01230 52957 36948 58846 03197 41673 16610 45426 66743 36816 69580 45119 23054 04890 66252 04399 12025 13268 13459 00000 00001 01065 46056 0.00556260 0.03207475 0.11262316 0.27565587 0.5132 5047 0. 77528358 147057 1.15799904 1. 24625601 1. 28758240 1.30364795 1.30891514 1.31037598 1.31072712 1.31079674 1.31080914 1.31061009 8.0 o.ooogoooo 00001 0053S 40745 0.00493716 0.02855264 052568 0.24663672 6018374 9633400 0.90070754 4249001 1.12271023 1. 16022587 1.17487526 1.17968845 1.18103027 1. 18134785 1. 16141270 1. 18142315 1.18142505 o.ggoooogo 00001 00839 36537 0.00443820 0.02572760 0.05077573 0. 22315073 1706765 3198936 0.81838596 0.94796127 2141762 5583477 6929874 737304 7 7456738 7526112 7532024 7533073 7533169 g.ogggoooo 00001 00759 33115 0.00403082 0.02341133 0.08275002 0.20374870 0.38134193 0.57853663 0.74936393 0.86916757 0.93691307 0.96870404 0.98115921 0.98526543 C.58641258 0.58668522 0.98674113 0.98675025 0.98675185

K. S. NAGARAJA AND G. R. VERMA 0.2 00000 00005 05453 0.00272363 0.03584117 0.21997245 0.81108451 6551933 3.97225761 6.15727425 8.11154556 9.50955105 10.3238466 10.7153769 10.8723373 10.9252405 10.9403435 10.9440088 10.9447689 10.9449053 10.9449253 O.OO00CO00 00003 04204 0.00221648 0.03029019 0.19067973 0. 71582013 1.84654099 3.58630848 5.59811974 7.41120720 8.71644688 9.48078346 5.84998512 9.99858185 488491 632334 66744e 674562 675898 676107 00000 00002 03406 0.00186903 0.02623474 0.16831416 4074504 1.67065334 3.26968002 5. 13375376 6.82468891 8.0487136e e.76e86654 9.11815262 9.25923341 9.30710506 9.32084274 9.32420731 9.32488823 9.32502365 5.32503851 0.8 00000 00002 02862 0.00161595 0.02314016 0. 15066642 0.58001363 1. 52534580 500870 4.74156761 6.32577601 7.47810841 8. 15891361 8.49032021 8.62460613 8.67029858 8.68344307 8.68667412 8.68733406 8.68745136 8.68747139 00000 00002 02469 0.00142334 0.02070025 0.13638043 C.52585144 1.40345478 2.78034306 4.40747452 5.85589214 6.98445588 7.62599630 7.94526577 8.07338142 8.11709118 8.12968522 8.13278770 6.13517857 8.13354301 8.13356209 0. 10 I. 5 00000 00001 01837 0.00109671 0.O1638466 0.110 26883 35 76062 1.17009163 2.34329033 3.74443817 4204464 5.99827862 6.569 89861 6.85079098 6.96583557 0525856 1667786 1950073 2008724 2019310 20 20931 00000 00001 01463 89209 0.01355956 0.09256206 0.37011248 348663 2551270 681114 4.40594006 5.25857639 5.77140427 6.02478405 6. 12909506 6.16499805 6.17543963 6.17802620 6.17857552 6.17867275 6.17868614 00000 00001 0104C 64971 0.01008376 0.07007289 0.28447515 0.78126515 1.59389687 2.58494755 3.52008515 4.22089005 4.64622402 4.85811424 4.94601059 4.97647361 4.58539543 4.98762226 4.98808861 4.58817444 4.98818588 00000 00000 00807 51052 0.00802675 38000 0.23104733 3972402 1.31417847 2.14344025 2.93178940 3.52665806 3.89000702 7207966 4. 14796629 4.17448711 4.18227196 4.18422508 4. 18463612 4.10470660 4.18472250 00000 00000 00659 42099 0.00666700 0.04716545 0.19452399 0.54163730 1.11808205 1.83056581 2.51236153 2516241 3.34625816 587749 3.57273293 3.55611797 3.60301971 3.60475349 3.60512257 3.60518837 3.6052007/ 6.0 U.000 0. 000 0.005 0.040 0.167 69 0.972 1.598 2.198 2.654 2.936 78 3. 138 3. 158 3.165 3.166 3.167 3. 167 3. 167 00000 ooooo 00557 35799 7U123 54044 97590 64645 95594 11306 10390 92630 24456 33672 04436 99066 19260 75854 08851 14764 16003 8:i 800000 ooooo 00482 31138 0.00497985 0.03554776 0.147e0605 1455531 0.86115837 1.41785336 1.95381832 2.36313629 2.61553056 2.74396515 2.79784296 2.81688023 2.82251356 2.82352883 2.82423306 2.82428932 2.82429655 d.o 8:88888888 8:8888388oü 00425 27552 0.00442060 0.03165005 0.13196162 0.37103641 0.77248216 1.27416611 845337 2.12921810 2.35873655 2.47524643 2.52437552 2.5417738C 2.54653222 2.54822922 2.54651151 2.54856014 2.54657159 9.0 00380 24706 0.00397426 0.02852275 0. 11518616 0.33578902 0.70034426 1. 15693 760 1.59863567 1.93748474 2.14765549 459862 960311 2.31571293 2.32046586 2. 32166862 2.32192707 2.32197666 2.32198238 Q.nooooooo 00000 00344 22394 0.00360579 0.02595799 866618 0.30665886 4053357 5947495 1.46546745 1.77746105 1.57129250 7007790 2.111 79256 2.12663555 2.13104916 2.13216305 2.13240623 2. 13244820 2.13245487

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