ABSTRACT
Following the methods of Chapter 4, we try to deform % in such a way that the absolute value of the integrand attains its maximum value on the contour a t w = In N. This condition is fulfilled by the contour indicated in Fig. 3.1, in which the radius R of the circular arc exceeds In N. This is because (w(-"-I obviously attains its maximum at w = In N, and
(exp(e")( 6 exp (ew( = exp(eRew) < 8, (3.03) equality being attained when w = In N. Letting R + ao we see that %' may be further deformed into the doubly infinite straight line Re w = In N and hence, by taking complex conjugates, that
1 In +tw exp (ew) a m = 1 m { , L N ?dW]*
compare (3.01). Thus M - )N as N - r a. We also have, again for large N, 1 1 -=-=-
1 {l + o(&)} = ={l +tO(l)}, w InN+it InN
uniformly with respect to t. Therefore
compare (3.01) and (3.04). t The reader will notice that the condition a < 1 of the theorem is violated. Provided that k is
