Evaluation of the Integral ftfu"e "\ u + x) ' du
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1 mathematics of computation volume 39, number 159 july 1982, pages 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, Mathematics Subject Classification. Primary 65A05, 65D20, 65D American Mathematical Society OO /82/O0OO-OO79/$O3.5O
2 180 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
3 EVALUATION OF THE INTEGRAL f^u"e~u\u + x) ' du 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 H 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 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.
4 182 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) <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
5 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 Department of Mathematics University of Rhode Island Kingston, Rhode Island M. Abramowitz, "Evaluation of the integral /0 e~"2'x/u du," J. Math. Phys., v. 32, 1953, pp A. Erdélyi, "Note on the paper 'On a definite integral' by R. H. Ritchie," MTAC, v. 4, 1950, p E. T. Goodwin & J. Staton, "Tables of f? e~u\u + x)~' du," Quart J. Mech. Appl. Math., v. 1, 1948, pp Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962, p K. S. Nagaraja, "Concerning the value of j{ e~" /(u + x) du," J. Math, and Phys., v. 44, 1965, pp F. W. J. Olver, Asymptotic Expansions and Special Functions, Academic Press, New York, R. H. Ritchie "On a definite integral," MTAC, v. 4, 1950, pp N. M. Temme, "Uniform asymptotic expansions of confluent hypergeometric functions," J. lnst. Math. Appl., v. 22, 1978, pp N. M. Temme, "The asymptotic expansions of the incomplete gamma functions," SI A M J. Math. Anal., v. 10, 1979, pp
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