Proofs

Most of the results in Section 2 are obtained by replacing an integrand by an approximation , writing , and finding upper and lower bounds for . All integrals are taken over the positive real line. The function is usually chosen to be a uniform approximation = + - , where is an approximation in the inner region, in the outer region, and in the overlap region or matching region. For instance, if (t) = [(t+x)(t+y)(t+z)]^-1/2 with x, y << z, we get by neglecting t compared to z, by neglecting x and y compared to t, and by doing both. A first example of this process is the proof of Lemma 1.

LEMMA 1.

If x 0, y 0, and 0 < x + y << z , then

where = g a = (x+ y)/2 with equalities if and only if x=y.

Proof. Let

Taking = + - , we find

and

Inequality (64) in the Appendix implies

and thus

.

As a second example, in which Lemma 1 is used, consider R_F (x, y, z) with x, y << z. Let

Taking = + - we find (with a and g the same as before)

and

Inequalities (61) and (64) imply

Hence, by Lemma 1,

where the last inequality follows from the next to last. We complete the proof of (26) by noting that

Equations (28) and (29) are obtained from [14, (2.15)(3.25),] with . To derive (32) we construct = + - as usual and find bounds for by using(60) and (65). To simplify the upper bound we note that R_F (x,y,z) R_F (x,y,0) and use (33).

Equations (22),(23),(24), and (25) follow from (32),(26),(29), and (27), respectively, by replacing x by y, replacing z by x, and simplifying.

Among the approximations for R_D we need discuss only (35) and (37), since (34), (36), (38), and (39) follow from (44), (47), (49), and (48), respectively, by putting p = z and simplifying. To prove (35) we let

choose = and apply (65) to get

Use of (22) completes the proof. Approximation (37) follows from applying (39) to two terms on the right side of

an identity that comes from [7, (5.9-5)(6.8-15),].

In discussing approximations for R_J we define

and construct , , and for each case in the manner described at the beginning of this Section. For example, if , then is obtained by neglecting t compared to p. Unless otherwise stated, we define (f_a = f_i + f_o - f_m ), take ( f_a ) as an approximation to ( f ), and find bounds for ( (f-f_a) ) by using the inequalities in the Appendix.

To prove (40) we use (69). To prove (41) we use (64) and note that ((f-f_a) = ( /p) f ). Before discussing (42), we consider (43), in which the error bounds are easily found by using (70). Finding ( f_a ) requires an integration by parts and a formula of which we omit the proof,

where ( = xy+ xz+yz ). To have a simpler approximation (), we define ( f_a = f_i + f_o - f_m ) and ( = f_i + f_s - f_m ), where ( f_o ) has been replaced by [ f_s(t) = 1t(t+g)^3/2 ] Then [ = 1xyz(4gp - 2), ] and an upper bound for ( (f-) ) is found by using ( (t+x)(t+y)(t+z) (t+g)^3/2 ) and (63). To find a lower bound, we note that ( f-f_a > 0 ), whence [ f- = f-f_a + f_o-f_s > f_o-f_s. ] A lower bound for ( (f_o-f_s) ) follows from (73).

The straightforward proof of (44) uses (64), (67), and Lemma 1. For the elementary approximation (45) we choose ( f_a =f_i ) and use (66). For the more accurate approximation (47) we take ( f_a = f_i + f_o - f_m ) and evaluate ( f_a ) by integrating by parts. The error bounds follow from (66) and (69) with two variables equated. To find the error bounds for (48), we use (68), (60), and (71) to prove [ t(t+x)(t+a) < 2gpx(f-f_a) < (ag + gp) 1t+z(t+g). ] After integration, (22) is used to complete the proof. In the case of (49), where ( (f_o - f_m) ) is infinite, we choose ( f_a = f_i ) and evaluate ( f_a ) by (20). It follows from (61) that [ 12x(t+y)(t+z)(1t+x - px(t+p)) < f_a - f < 12x(t+x)(t+y)(t+z), ] where we have replaced ( t/(t+p) ) by ( 1 ) in the upper bound and ( x/(x-p) ) by 1 in the lower bound. We then use (20), (55), and (27).

The function can be expressed in terms of and by (17) and [7, Table 9.3-1,]:

Applying (26) and (34), we obtain (51). The error bounds have been substantially simplified by using the numerical value of and assuming (5a < z ) in the upper bound. It is not hard to obtain (53) from a well-known infinite series [15, p. 54,] for by using the inequality [ 1 + 38k'^2 < , _2 F_1(12 ,32 ;2;k'^2) < (1-k'^2)^-1/2 = 1/k, 0 < k' < 1, ] for the hypergeometric function . Unfortunately (58) does not lead to simple error bounds for (54). Instead, we define ( f(z) = R_G(x,y,z) ) and find from [7, (5.9-9)(6.8-6),] that [ f'(z) = 180^tdt(t+x)(t+y)(t+z)^3/2. ] Since this is a strictly decreasing function of z, the mean value theorem yields ( f(z) = f(0) + zf'() ) where [ f'(z) < f'() < f'(0) = 14R_F(x,y,0). ] By (71) and (5) we see that [ f'(z) 18 0^tdt(t+a)(t+z)^3/2 = 14(a-z)[-z + aR_C(z,a)]. ] Use of (33) and (22) completes the proof of (54).