ABSTRACT
In (7.06) and (7.07) the upper or lower signs are taken according as x >< xo. The auxiliary functions E(1;), M ( 0 , and the constant 1 = 1.04 ... are defined in Chapter 11, $2.
Hypothesis (iii) implies that if x -+ b then C -+ + m, and if x -+ a then 1; -+ - m. On substituting for Ai(1;) and the auxiliary functions by means of their asymptotic forms for large arguments (Chapter 11, 551,2), we obtain
57 The Gans-Jeffreys Formulas: Real-Variable Method
where q4(x) = n'I2 JC(114~4(~), and is bounded. Comparison of (7.03) and (7.08) immediately reveals that
Next, because the general solution of (7.01) can be represented by Aw,(B,x), where A and 8 are disposable constants, we can express
(7.10) where a and 6 are independent of x. Comparing (7.09) with (7.10) and following the steps of Chapter 6, $7.3, we derive
provided that the right-hand side of this inequality does not exceed unity. Therefore
This is the first Gans-Jeflreys formula: its error terms a and 6 are bounded by (7.1 1). If, for example, f contains a multiplicative parameter uZ, then both a and 6 are ~ ( u - ' ) as u -, oo. 7.3 In a similar way, we find that
w5(x) = n-11y-114{l + b ( ~ ) + ~ ( l ) } exp([ f llzdt) (x-, b), (7.13)
and
tt5 (x) = ~ ~ ~ ~ C ~ ~ ~ e x p ( - 3 ~ ~ ~ ~ ) ~ ~ (x). Comparison of (7.05) and (7.14) shows that
Also, if x -, b then q5(x) tends to a finite limit q(b), say; compare Chapter 11, Exercise 4.1. Therefore we have
in the sense that there is a particular solution wz(x) with the property (7.04) which satisfies this equation. This is the second Gans-Jeffreys formula. Its error term is bounded by
