BOUNDARY PROBLEMS AND DEVELOPMENTS. 83 



(43) U(x) = Six) CT(x) - — / Six) T(t)*(t, X) U(t)S(t) T(x) dt. 



°-r»P ( 



Theorem: If the functions 74(0*), i — 1, 2, . . .11, satisfy the relations 

 arg iy jix) — 7i(a*)} = hi,-, i, j, = 1,2,. . .n, where each hij is a con- 

 stant, then there corresponds to each sector bounded by two adjacent 

 rays R\\{yj(x) — ji(x)} } — a choice of the lower limits of integra- 

 tion which is such that for X within the sector and | X | > N, and for 

 any continuous matrix U(t), each element of the matrix 



X 



yp{x, X) = Cs(x) f{t) $it, X) Uit) Sit) Tix) dt 



is less numerically than KM, where M is the largest numerical maxi- 

 mum attained by any element of U. 

 Proof: Writing 



Six) = isvix)) Eix), fix) = E-^ix) (%(*)), 

 we have 



+(*, X) = ( I fs ih ix)^ rh{x) - Thil)l t hl it) «, lp (t, X) u vq it) 



\ h, l, p, a, r-l J 



s gr it)eMrrW-rr(x)\ tr .^ dt \ 



n 1 x 



x/(7 A a)-7 r «)}di (hr) 



<*"%> it, x, x) dt/, 



where, for | X | > N, | to ( {, r) | < /c ( * r) M for all ?' and j, /c ( * r) being a 

 positive constant. 



Consider a particular sector and any element 



X 



X/{7 A «)-7 r (f)}d{ 



ffr. 



If #|X{7/,(£) — 7 r (^)| ^ for X within the sector and any £ it is so 

 for all £, and, provided that t ^ x, the integral 



8 The notation R \<p\ is used to indicate " the real part of <p." 



