Asymptotic Expansions of a Class of Integrals

A. Sellier
1994 Proceedings of the Royal Society A  
T h e p u rp o se of th is p a p e r is to stu d y th e expansion w ith resp ect to th e large and real p a ra m e te r A of th e in teg ral /(A ) := fp f f(x Jo w hen w eak assu m p tio n s are req u ired for th e pseudofunctions / ( an d an d th e sym bol fp m eans th e in teg ratio n in th e finite p a rt sense of H ad am ard . J(X):=(H(X x )J (x )),(1.3) w here ( , ) denotes th e d u ality bracket and / belongs to a correct space of test functions or pseudofunctions. f T h is paper w as
more » ... h is paper w as produ ced from th e au th o r's disk by using th e T^X ty p e se ttin g system . Proc. R . Soc. Loud. A (1994) 4 4 5 , 6 9 3 -7 1 0 Printed in Great Britain 693 (c) 1994 T he Royal Society P r o p o s i t i o n 1. Given two real values c and d with 0 -< then for a e C and j e N d J x< * logJ xdx = ) -P3 a(c with else7 W = ^E^; ! f l t l0gfe, k!(a + i y + i -kl og W* (2-4) Proc. R. Soc. Lond. A (1994) J c J c/A sgn(A)/e /(A t)d (A t) pl/e J |A|/e' / ( x ) d x ( 2.20) m,Re (7) ^l-bRe(ai)
doi:10.1098/rspa.1994.0087 fatcat:al2nphc43fgrrlvjzejban2ac4