ABSTRACT
The effects of this extension on the example depicted in Figs. 9.2-9.7 are as follows. First, the shadow zone GHCB associated with Z,(u) can be reduced to the cut BC by taking r large enough. Secondly, the arcs DIand JK bounding the shadow zones FIDE and BKJA of Fig. 9.6 are replaced by concentric arcs with radius increased by the factor r. In consequence, by taking r large enough we can extend Z,(u, me-"") to include any point of the original shadow zones. It should be noticed, however, that R(r) -+ m as r -+ co, implying that the error term E,,+ ,, , (u, t;) may become prohibitively large. 10.3 A second extension of the regions of validity is to include sheets of the t; plane beyond the sector Jph(~t;"~)1< fn. This is easily accomplished in the case of W2n+l,l(~,t;), as follows. If t; lies in the sector
m denoting any integer, then we set % = t;ezmni, so that Jph(u1112)1 < fn. The integral equation (3.08) for the error term may be set up with [ in place oft; and solved in the same way. On restoring the original variables, we see that the expansion (9.02) holds when t; E Z,(u) for any value of ph C, provided that the branch of ['I2 in (9.04) and (9.05) is chosen to satisfy Iph(ut;'I2)1< fn.
