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y Fm(n)798 1538 y Fl(p)p 836 1538 78 2 v 38 x Ft(2)p Fn(\031)r(n)918 1540 y Ft(\(1)e(+)g Fn(o)p Ft(\(1\)\))p Fn(;)53 b(n)12 b Fl(!)i(1)p Fn(:)-54 1660 y Ft(The)h(pro)q(of)g(strategy)f(is)i(as)f(follo)o(ws:)2 1757 y(1.)22 b(\014rst)14 b(pro)o(v)o(e)g(the)h(existence)h(of)e Fn(\032)g Ft(b)o(y)g(an)h(in)o(termediate)g(v)m(alue)h(argumen)o(t)e(and)h (compute)g(an)f(asymptotic)60 1813 y(estimate)h(b)o(y)g(in)o(v)o(ersion:)728 1893 y Fn(\032)d Ft(=)823 1863 y Fn(f)5 b Ft(\(0\))p 817 1883 97 2 v 817 1924 a Fn(f)844 1911 y Fi(0)856 1924 y Ft(\(0\))924 1863 y Fn(n)p 924 1883 28 2 v 926 1924 a(d)956 1893 y Ft(\(1)10 b(+)g Fn(O)q Ft(\()p Fn(n=d)p Ft(\)\);)2 2031 y(2.)22 b(from)14 b(this)i(deduce)g(an)f(estimate)g(for)g Fn(f)749 2014 y Fq(\()p Fm(k)q Fq(\))798 2031 y Ft(\()p Fn(\032)p Ft(\))c(=)i Fn(f)944 2014 y Fq(\()p Fm(k)q Fq(\))993 2031 y Ft(\(0\))d(+)g Fn(O)q Ft(\()p Fn(n=d)p Ft(\))k(and)247 2166 y Fn(\032)271 2148 y Fq(2)291 2166 y Fn(h)317 2148 y Fi(00)338 2166 y Ft(\()p Fn(\032)p 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Ft(\()p Fn(z)r Ft(\))p Fn( )r Ft(\()p Fn(z)r Ft(\))f(with)i(the)g (same)f(h)o(yp)q(othesis)h(on)g Fn(f)20 b Ft(and)15 b Fn(d)f Ft(and)-54 2588 y(a)h(function)h Fn( )g Ft(analytic)g(and)f(non-zero)g(at)g (the)g(origin,)g(to)f(b)q(e)i(sp)q(eci\014ed.)22 b(In)16 b(this)f(case,)g(it) g(is)h(kno)o(wn)f(that)f(the)-54 2645 y(saddle-p)q(oin)o(t)19 b(metho)q(d)e(still)h(w)o(orks)e(with)h(the)g(same)g(saddle-p)q(oin)o(t)h (and)f(the)g(same)g(function)g Fn(\017)p Ft(\()p Fn(n)p Ft(\))g(as)g(b)q (efore)-54 2701 y(pro)o(vided)f(that)p -54 2743 780 2 v -2 2770 a Fo(5)15 2786 y Fx(As)d(it)g(is)h(stated,)f(Go)q(o)q(d's)h(theorem)f (\(p.868\))g(is)h(wrong,)f(a)g(coun)o(ter-example)i(b)q(eing)f([)p Fd(z)1306 2770 y Ff(n)1327 2786 y Fx(]\(1)8 b(+)h Fd(z)r Fx(\))1454 2770 y Ff(n)1474 2786 y Fx(.)p eop %%Page: 86 4 86 3 bop -54 99 a Fu(86)1365 b Fc(P)o(art)14 b(I)q(I)q(I.)f(Asymptotic)g (Analysis)p -54 118 1950 2 v -37 278 a Fp(\(d.\))22 b Fl(j)p Fn( )r Ft(\()p Fn(\032e)168 261 y Fm(i\022)197 278 y Ft(\))p Fl(j)12 b(\030)h Fn( )r Ft(\()p Fn(\032)p Ft(\))h(uniformly)i(for)e(0)f Fl(\024)g(j)p Fn(\022)q Fl(j)f(\024)h Fn(\017)j Ft(or;)-51 374 y Fp(\(d'.\))22 b Ft(uniformly)16 b(for)e(0)f Fl(\024)g(j)p Fn(\022)q Fl(j)f(\024)h Fn(\017)p Ft(,)i(the)h(follo)o(wing)g(estimates)f (hold:)435 450 y Fj(\014)435 475 y(\014)435 499 y(\014)435 524 y(\014)454 480 y Fn( )486 464 y Fi(0)p 454 500 43 2 v 460 542 a Fn( )502 450 y Fj(\014)502 475 y(\014)502 499 y(\014)502 524 y(\014)523 511 y Ft(\()p Fn(\032)p Ft(\))d(=)h Fn(o)p Ft(\()683 460 y Fj(q)p 724 460 107 2 v 724 511 a Fn(h)750 498 y Fi(00)771 511 y Ft(\()p Fn(\032)p Ft(\))o(\))p Fn(;)959 437 y Fj(\014)959 462 y(\014)959 487 y(\014)959 512 y(\014)959 537 y(\014)973 451 y(\022)1008 480 y Fn( )1040 464 y Fi(0)p 1008 500 43 2 v 1014 542 a Fn( )1056 451 y Fj(\023)1087 458 y Fi(0)1098 437 y Fj(\014)1098 462 y(\014)1098 487 y(\014)1098 512 y(\014)1098 537 y(\014)1120 511 y Ft(\()p Fn(\032e)1183 492 y Fm(i\022)1214 511 y Ft(\))f(=)h Fn(o)p Ft(\()p Fn(h)1358 492 y Fi(00)1379 511 y Ft(\()p Fn(\032e)1442 492 y Fm(i\022)1473 511 y Ft(\)\);)-35 673 y Fp(\(e.\))22 b Fl(j)p Fn( )r Ft(\()p Fn(\032e)168 656 y Fm(i\022)197 673 y Ft(\))p Fl(j)12 b Ft(=)h Fn(O)q Ft(\()p Fn( )r Ft(\()p Fn(\032)p Ft(\)\))g(uniformly)j(for)f Fn(\017)e(<)g Fl(j)p Fn(\022)q Fl(j)f Fn(<)h(\031)r Ft(;)-54 769 y(and)20 b(then)g(the)g(asymptotic)f(estimate)h(has)f(to)g(b)q(e)h(m)o(ultiplied)j(b)o (y)d Fn( )r Ft(\()p Fn(\032)p Ft(\).)31 b(Conditions)20 b Fp(\(d.\))35 b Ft(and)20 b Fp(\(e.\))34 b Ft(are)-54 826 y(ob)o(viously)14 b(ful\014lled)j(if)d Fn( )g Ft(is)g(an)g(analytic)g(function)g(whic)o(h)h(do) q(es)f(not)f(dep)q(end)i(on)e Fn(d)h Ft(or)f Fn(n)p Ft(,)g(but)h(more)f (generally)l(,)-54 882 y(D.)i(Gardy)f(considers)i(functions)g Fn( )h Ft(satisfying)e Fr(H1)g Ft(and)h(of)e(the)i(form)577 1017 y Fn( )r Ft(\()p Fn(z)r Ft(\))11 b(=)747 961 y Fm(p)729 976 y Fj(Y)727 1068 y Fm(i)p Fq(=1)792 1017 y Fn(g)814 1024 y Fm(i)828 1017 y Ft(\()p Fn(z)r Ft(\))887 998 y Fm(d)905 1003 y Ff(i)920 1017 y Fn(;)52 b(d)1009 1024 y Fm(i)1036 1017 y Ft(=)13 b Fn(o)p Ft(\()p Fn(d=)1171 982 y Fl(p)p 1208 982 28 2 v 1208 1017 a Fn(n)p Ft(\))p Fn(:)-54 1158 y Ft(Since)20 b Fn( )r Ft(\(0\))d Fl(6)p Ft(=)i(0,)g(it)f(is)i(easy)e(to)g(see)h(that)f Fp(\(d'.\))32 b Ft(is)19 b(then)g(satis\014ed,)h(and)e Fp(\(e.\))32 b Ft(follo)o(ws)18 b(immediately)j(from)-54 1214 y(Lemma)15 b(1.)-54 1338 y Fr(Note.)45 b Ft(The)16 b(su\016cien)o(t)g(conditions)g(used) g(in)g(this)g(summary)e(are)h(tak)o(en)g(or)f(deriv)o(ed)j(from)d([1)o(].)-54 1478 y Fv(References)-33 1570 y Fu([1])21 b(L.)12 b(Berg.)18 b(Asymptotics)12 b(of)f(in)o(tegrals,)h(series,)i(and)e(op)q(erators.)18 b(In)12 b(R.)g(W)m(ong,)e(editor,)j Fg(Asymptotic)g(and)h(Computa-)33 1620 y(tional)g(A)o(nalysis)p Fu(,)e(v)o(olume)f(124)h(of)h Fg(L)n(e)n(ctur)n(e)g(notes)h(in)g(pur)n(e)g(and)h(applie)n(d)f(mathematics)p Fu(,)e(pages)i(35{52.)d(M.)h(Dekk)o(er,)33 1670 y(New)j(Y)m(ork,)e(Basel,)h (1990.)j(Conference)e(in)e(Honor)h(of)f(Frank)h(W.)f(J.)h(Olv)o(er's)g(65th)g (Birthda)o(y)m(.)-33 1760 y([2])21 b(H.)12 b(E.)g(Daniels.)k(Saddlep)q(oin)o (t)c(appro)o(ximations)d(in)j(statistics.)18 b Fg(A)o(nnals)c(of)f (Mathematic)n(al)g(Statistics)p Fu(,)e(25:631{650,)33 1810 y(1954.)-33 1899 y([3])21 b(D.)11 b(Gardy)m(.)16 b(On)c(co)q(e\016cien)o(ts)h (of)e(p)q(o)o(w)o(ers)h(of)f(functions.)17 b(T)m(ec)o(hnical)11 b(Rep)q(ort)h(CS-91-53,)f(Bro)o(wn)g(Univ)o(ersit)o(y)m(,)h(August)33 1949 y(1991.)-33 2039 y([4])21 b(D.)10 b(Gardy)m(.)15 b(Some)9 b(results)i(on)f(the)h(asymptotic)e(baha)o(viour)g(of)g(co)q(e\016cien)o(ts)i (of)f(large)f(p)q(o)o(w)o(ers)i(of)f(functions.)16 b(Rapp)q(ort)33 2089 y(de)f(rec)o(herc)o(he)h(769,)d(Lab)q(oratoire)g(de)i(rec)o(herc)o(he)h (en)f(informatique,)c(Univ)o(ersit)o(\023)-20 b(e)14 b(de)g(P)o(aris)g(Sud,)g (August)g(1992.)-33 2178 y([5])21 b(I.)14 b(J.)g(Go)q(o)q(d.)k(Saddle-p)q (oin)o(t)c(metho)q(ds)f(for)h(the)h(m)o(ultinom)o(ia)o(l)c(distribution.)18 b Fg(A)o(nnals)e(of)f(Mathematic)n(al)g(Statistics)p Fu(,)33 2228 y(28:861{881,)c(1957.)-33 2318 y([6])21 b(B.)15 b(Harris)h(and)f(L.)g (Sc)o(ho)q(enfeld.)22 b(Asymptotic)14 b(expansions)i(for)f(the)g(co)q (e\016cien)o(ts)i(of)d(analytic)h(functions.)22 b Fg(Il)r(linois)33 2368 y(Journal)15 b(of)g(Mathematics)p Fu(,)f(12:264{277,)c(1968.)-33 2457 y([7])21 b(W.)15 b(K.)h(Ha)o(yman.)22 b(Co)q(e\016cien)o(t)16 b(problems)f(for)h(univ)n(alen)o(t)f(functions)g(and)h(related)h(function)e (classes.)25 b Fg(Journal)17 b(of)33 2507 y(the)e(L)n(ondon)h(Mathematic)n (al)f(So)n(ciety)p Fu(,)f(40:385{406,)d(1965.)-33 2597 y([8])21 b(A.)d(Meir)g(and)g(J.)g(W.)f(Mo)q(on.)30 b(The)19 b(asymptotic)d(b)q(eha)o (viour)i(of)f(co)q(e\016cien)o(ts)j(of)d(p)q(o)o(w)o(ers)i(of)e(certain)i (generating)33 2647 y(functions.)f Fg(Eur)n(op)n(e)n(an)e(Journal)f(of)g (Combinatorics)p Fu(,)d(11:581{587,)f(1990.)-33 2736 y([9])21 b(A.)14 b(M.)g(Odlyzk)o(o)g(and)g(L.)g(B.)g(Ric)o(hmond.)j(Asymptotic)d (expansions)g(for)g(the)h(co)q(e\016cien)o(ts)h(of)d(analytic)h(generating)33 2786 y(functions.)k Fg(A)n(e)n(quationes)e(Mathematic)n(ae)p Fu(,)e(28:50{63,)d(1985.)p eop %%Page: 87 5 87 4 bop -54 98 a Fh(The)15 b(Asymptotic)h(Beha)o(viour)f(of)g(Co)q (e\016cien)o(ts)h(of)f(Large)g(P)o(o)o(w)o(ers)e(of)i(F)l(unctions)479 b Ft(87)p -54 118 1950 2 v -54 278 a Fu([10])21 b(L.)14 b(Siro)o(vic)o(h.)j Fg(T)m(e)n(chniques)e(of)g(Asymptotic)f(A)o(nalysis)p Fu(.)k(Springer)c(V)m (erlag,)f(1971.)-54 365 y([11])21 b(J.-M.)13 b(Stey)o(aert)h(and)f(Ph.)f(Fla) r(jolet.)17 b(P)o(atterns)e(and)e(pattern-matc)o(hing)e(in)i(trees:)19 b(an)13 b(analysis.)k Fg(Information)d(and)33 415 y(Contr)n(ol)p Fu(,)f(58\(1{3\):19{58,)e(July)i(1983.)-54 502 y([12])21 b(M.)15 b(Wyman.)21 b(The)16 b(asymptotic)e(b)q(eha)o(vior)h(of)g(the)h(Lauren)o(t)g (co)q(e\016cien)o(ts.)24 b Fg(Canadian)17 b(Journal)f(of)g(Mathematics)p Fu(,)33 552 y(11:534{555,)11 b(1959.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF