By Maurice Holt

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**Example text**

Exercises 1. 38. 2. The solutions of the differential equation u" + [1 - 21/[1 - L(L + 1)x- 2 ]u = 0, x> 0, L~-t are called Coulomb wave functions; they are closely related to confluent hypergeometric functions [44]. Show that the solution F L (1/, x) which has the limit as x ~ 0+ (the one commonly used in applications) is zeromaximal in (0, ex). 3. Show that the kth positive zero of the hypergeometric function F(rx, p, y, x) [44] (if it exists) exceeds the kth positive zero of F(rx - y + 1, p - y + 1, 2 - y, x) if y > 1, y "# 2, 3, ....

Fl· 46 2 OSCILLATION AND NONOSCILLATION THEOREMS In particular this shows that g* ~ ! 4) is not finite. 3) that, for given e > 0, there is a number Xo > 0 such that IC(X) _ xw I< ~x 2 2 provided x > Xo. ] [C(t) ~! = e. 1 shows that Eq. 1) is oscillatory if w >! and nonoscillatory if w < 1. 1) and the eigenvalue problem U" + AC(X)U = 0, U(IX) = u'(f3) = o. 54) are replaced by 1, 0, c(x), U(IX) , and u'(f3), respectively. 51) becomes i .. [uJ f.. U,2 dx, /l = where 1).. = {u E ~1[1X, 13]: U(IX) = o}.

14) of Section 1. 8.