ABSTRACT
Equation (7.02) has the same form as (6.03) of Chapter l l and may be analyzed in the same way. We first observe from 51.3 that if u J C ~ ' / ~ <y , , , then we have -an < B,(U)C~'/~) < 0 and COSB, (U~C~~/~ ) 2 2-'12. Since
JP~(u,C)I < L3(v)g(u) < LZ(V)O(U) < 2-'12. (7.03) it follows that there can be no zeros in the C interval [-y:, ,/u2,0). 7.2 For the C interval ( - a , -y:, ,/u2), we find from (7.02)
B , (u~[ ( '~~ ) - (n-*)IT + (-)"sin-'{p3(u,0} = 0, where n is an arbitrary positive integer. With u > u, the principal value of the inverse sine is numerically less than an. Hence from the first of (1.21) we see that there is at least one zero in the interval
To delimit this zero in a shorter interval, we have
compare (1.20) and (4.03). Also, in consequence of (7.03) we have
Combination of these results shows that, for each positive integer n, W,,,(u,C) vanishes for at least one value of C satisfying
Asymptotically,
as u -r m, uniformly for those values of n for which y,,+, < u lal'/2. If a = - m, then all values of n are included.
