. The Bell System technical journal . t;^), etc.,as functions of m = n -\- I than as functions of z. In the eailier sections E. T. Boll, Kxj)()iipiiti:il Polynomials, .Vnii. of 35, pp. 25S-27y, 1> ^ .^^. BOUNDARY BETWEEN to HALF-PLANE AND t, HALF-PLANE Fig. — Diagram showing the half-plane regions to which the saddle pointsto and ti are confined in the /-plane. One might wonder why cuts in the w-plane are required since it hasalready been pointed out that Un(z), etc., are one-valued functions ofm = n + 1. The trouble is that the asymptotic expressions for Un{z)are many-valued fu
. The Bell System technical journal . t;^), etc.,as functions of m = n -\- I than as functions of z. In the eailier sections E. T. Boll, Kxj)()iipiiti:il Polynomials, .Vnii. of 35, pp. 25S-27y, 1> ^ .^^. BOUNDARY BETWEEN to HALF-PLANE AND t, HALF-PLANE Fig. — Diagram showing the half-plane regions to which the saddle pointsto and ti are confined in the /-plane. One might wonder why cuts in the w-plane are required since it hasalready been pointed out that Un(z), etc., are one-valued functions ofm = n + 1. The trouble is that the asymptotic expressions for Un{z)are many-valued functions of m even though Un{z) itself is not. Now that we have considered the saddle points ta and /i, we turn toa consideration of the paths of steepest descent in the /-plane whichpass through them.* The path of steepest descent which passes through/o, for example, is that branch of the curve Im im - f(U)] = 0 () for which to is the highest point (, Re [f(t) - f(to)] ^ 0 on it). The * Watson^s has studied paths corresponding to Re(n) > 0 when z is any com-plex number, and has given curves which are related to some of those shown inSection 11. DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDE
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