This is a multi-part message in MIME format. ---------------0406230407465 Content-Type: text/plain; name="04-195.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-195.keywords" partial differential equations, analytic continuation,power series solutions, summability ---------------0406230407465 Content-Type: application/postscript; name="holcont.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="holcont.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: holcont.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o holcont.ps holcont.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.06.08:1619 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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b(l)29 b Fn(z)h Fd(2)24 b Fn(D)s Fo(\(0)p Fn(;)15 b(\032)p Fo(\))27 b Fd(\032)e Fo(\012)p Fh(,)k(the)g(series)f Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\))30 b Fh(c)-5 b(an)29 b(b)-5 b(e)28 b(analytic)-5 b(al)5 b(ly)75 2870 y(c)-5 b(ontinue)g(d)37 b(with)g(r)-5 b(esp)g(e)g(ct)37 b(to)f Fn(t)g Fh(in)g(a)g(se)-5 b(ctor)37 b Fn(S)1688 2885 y Fm(d;\016)1812 2870 y Fo(=)31 b Fd(f)p Fn(t)g Fd(2)g Fl(C)55 b Fo(:)31 b Fd(j)p Fn(d)23 b Fd(\000)f Fo(arg)r(\()p Fn(t)p Fo(\))p Fd(j)32 b Fn(<)e(\016)s Fd(g)p Fh(.)53 b(Mor)-5 b(e)g(over,)38 b(ther)-5 b(e)36 b(exist)75 2983 y Fn(\016)115 2997 y Fe(1)180 2983 y Fn(<)25 b(\016)s Fh(,)33 b Fn(C)f(>)25 b Fo(0)p Fh(,)33 b(and)g Fn(M)j(>)25 b Fo(0)33 b Fh(such)f(that)1511 3196 y Fo(sup)1435 3284 y Fm(z)s Fg(2)1534 3267 y Fe(\026)1518 3284 y Fm(D)q Fe(\(0)p Fm(;\032)p Fe(\))1739 3196 y Fd(j)p Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\))p Fd(j)26 b(\024)f Fn(C)7 b(e)2313 3159 y Fm(M)g Fg(j)p Fm(t)p Fg(j)2453 3135 y Fc(k)75 3472 y Fh(for)33 b(al)5 b(l)33 b Fn(t)25 b Fd(2)g Fn(S)550 3487 y Fm(d;\016)637 3496 y Fb(1)676 3472 y Fh(.)75 3685 y Fo(If)30 b(this)f(is)h(so,)g(the)h(Laplace)g (transform)e(of)i(order)f Fn(k)j Fo(of)e Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\),)75 3955 y(\()p Fn(X)7 b Fo(\))950 b Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)e Fn(t)1604 3917 y Fg(\000)p Fm(k)1716 3831 y Ff(Z)1807 3857 y Fg(1)p Fe(\()p Fm(\015)t Fe(\))1767 4037 y(0)1992 3955 y Fn(W)13 b Fo(\()p Fn(\034)5 b(;)15 b(z)t Fo(\))p Fn(e)2334 3917 y Fg(\000)p Fe(\()p Fm(\034)8 b(=t)p Fe(\))2542 3894 y Fc(k)2587 3955 y Fn(d\034)2684 3917 y Fm(k)2727 3955 y Fn(;)75 4205 y Fo(where)31 b(the)g(in)m(tegration)g(is)f(made)h (along)g(the)g(ra)m(y)h(arg)r(\()p Fn(\034)10 b Fo(\))27 b(=)f Fn(\015)36 b Fo(with)30 b Fd(j)p Fn(d)21 b Fd(\000)f Fn(\015)5 b Fd(j)27 b Fn(<)f(\016)s Fo(,)32 b(is)e(called)g(the)i Fn(k)s Fo(-sum)e(of)75 4318 y(the)h(formal)e(series)787 4295 y(^)761 4318 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\).)75 4480 y(Rewrite)30 b(the)g(formal)g(series)g(\()1162 4457 y(^)1136 4480 y Fn(X)8 b Fo(\))30 b(in)f(a)i(formal)f(series)f(with)g (resp)s(ect)i(to)g(the)f(v)-5 b(ariable)30 b Fn(z)t Fo(,)1538 4727 y(^)1512 4750 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1906 4664 y Ff(X)1906 4860 y Fm(n)p Fg(\025)p Fe(0)2090 4666 y Fo(^)2064 4689 y Fn(X)2139 4703 y Fm(n)2186 4689 y Fo(\()p Fn(t)p Fo(\))p 2064 4729 226 4 v 2137 4813 a Fn(n)p Fo(!)2299 4750 y Fn(z)2345 4713 y Fm(n)2393 4750 y Fn(;)75 5066 y Fo(where)364 5043 y(^)338 5066 y Fn(X)413 5080 y Fm(n)460 5066 y Fo(\()p Fn(t)p Fo(\),)k Fn(n)25 b Fd(\025)g Fo(0,)31 b(are)g(no)m(w)f(formal)g(series.)40 b(W)-8 b(e)31 b(ha)m(v)m(e)h(the)f(follo)m(wing)d(result)i(\(see)h ([9]\),)75 5254 y Fi(Prop)s(osition)36 b(1)46 b Fh(Consider)34 b(the)f(line)-5 b(ar)33 b(p)-5 b(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation,)75 5466 y Fo(\()p Fn(E)177 5480 y Fm(P)10 b(V)293 5466 y Fo(\))692 b Fn(@)1073 5429 y Fm(S)1068 5489 y(z)1150 5443 y Fo(^)1124 5466 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1623 5380 y Ff(X)1518 5581 y Fe(\()p Fm(k)1582 5590 y Fb(0)1617 5581 y Fm(;k)1674 5590 y Fb(1)1708 5581 y Fe(\))p Fg(2S)1825 5590 y Fb(1)1874 5466 y Fn(A)1942 5485 y Fe(\()p Fm(k)2006 5494 y Fb(0)2041 5485 y Fm(;k)2098 5494 y Fb(1)2132 5485 y Fe(\))2164 5466 y Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(@)2406 5426 y Fm(k)2443 5435 y Fb(0)2401 5490 y Fm(t)2483 5466 y Fn(@)2536 5429 y Fm(k)2573 5438 y Fb(1)2531 5489 y Fm(z)2638 5443 y Fo(^)2612 5466 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(;)p eop %%Page: 3 3 3 2 bop 3809 100 a Fo(3)75 399 y Fh(wher)-5 b(e)35 b Fd(S)388 413 y Fe(1)461 399 y Fh(is)f(a)g(\014nite)g(subset)f(of)h Fl(N)1314 366 y Fe(2)1393 399 y Fh(and)h Fn(A)1639 417 y Fe(\()p Fm(k)1703 426 y Fb(0)1738 417 y Fm(;k)1795 426 y Fb(1)1829 417 y Fe(\))1860 399 y Fo(\()p Fn(t;)15 b(z)t Fo(\))35 b Fh(ar)-5 b(e)35 b(holomorphic)i(functions)d(in)g(a)g (neighb)-5 b(orho)g(o)g(d)75 528 y(of)31 b(the)h(origin)f(in)g Fl(C)764 495 y Fe(2)809 528 y Fh(.)41 b(L)-5 b(et)32 b Fo(\()1095 505 y(^)1069 528 y Fn(X)8 b Fo(\))31 b Fh(b)-5 b(e)31 b(a)g(formal)i(solution)f(of)g Fo(\()p Fn(E)2243 542 y Fm(P)10 b(V)2359 528 y Fo(\))p Fh(.)41 b(Assume)31 b(that)h(ther)-5 b(e)32 b(exists)g(an)f(inte)-5 b(ger)75 641 y Fn(K)32 b(>)25 b Fo(1)33 b Fh(such)g(that)906 754 y Fd(S)961 768 y Fe(1)1026 754 y Fo(=)25 b Fd(f)p Fo(\()p Fn(k)1249 768 y Fe(0)1289 754 y Fn(;)15 b(k)1376 768 y Fe(1)1416 754 y Fo(\))26 b Fd(2)f Fl(N)1622 716 y Fe(2)1693 754 y Fo(:)h Fn(k)1791 768 y Fe(0)1830 754 y Fn(K)h Fo(+)20 b Fn(k)2072 768 y Fe(1)2137 754 y Fd(\024)25 b Fn(S;)15 b Fo(\()p Fn(k)2411 768 y Fe(0)2452 754 y Fn(;)g(k)2539 768 y Fe(1)2579 754 y Fo(\))25 b Fd(6)p Fo(=)g(\(0)p Fn(;)15 b(S)5 b Fo(\))p Fd(g)p Fn(:)75 903 y Fh(We)27 b(assume)h(that)g(the)g(formal)h(series)1435 880 y Fo(^)1409 903 y Fn(X)1484 917 y Fm(\026)1531 903 y Fo(\()p Fn(t)p Fo(\))p Fh(,)g(for)e Fo(0)f Fd(\024)f Fn(\026)g Fd(\024)g Fn(S)14 b Fd(\000)9 b Fo(1)p Fh(,)27 b(ar)-5 b(e)29 b Fo(1)p Fn(=)p Fo(\()p Fn(K)15 b Fd(\000)9 b Fo(1\))p Fh(-summable)29 b(with)f(r)-5 b(esp)g(e)g(ct)75 1016 y(to)33 b Fn(t)g Fh(in)f(a)h(dir)-5 b(e)g(ction)34 b Fn(d)26 b Fd(2)f Fl(R)r Fh(.)216 1129 y(Mor)-5 b(e)g(over,)51 b(we)46 b(assume)h(that)h(their)f(sums)f Fn(X)1890 1143 y Fm(\026)1937 1129 y Fo(\()p Fn(t)p Fo(\))h Fh(ar)-5 b(e)47 b(holomorphic)i(in)e(a)f(se)-5 b(ctorial)48 b(r)-5 b(e)g(gion)47 b Fn(G)k Fd(\032)75 1242 y Fn(D)s Fo(\(0)p Fn(;)15 b(R)q Fo(\()p Fn(\032)p Fo(\)\))p Fh(,)31 b(indep)-5 b(endent)31 b(of)f Fn(\026)p Fh(,)f(which)h(has)h(bise)-5 b(cting)29 b(dir)-5 b(e)g(ction)30 b Fn(d)g Fh(and)g(op)-5 b(ening)30 b(gr)-5 b(e)g(ater)31 b(than)f Fo(\()p Fn(K)20 b Fd(\000)13 b Fo(1\))p Fn(\031)75 1355 y Fh(and)34 b(satisfy)f(the)g (fol)5 b(lowing)34 b(estimates.)75 1468 y(F)-7 b(or)34 b(al)5 b(l)33 b(close)-5 b(d)34 b(se)-5 b(ctor)919 1445 y Fo(\026)903 1468 y Fn(S)31 b Fd(\032)25 b Fn(G)p Fh(,)32 b(ther)-5 b(e)34 b(exist)f Fn(C)q(;)1787 1445 y Fo(~)1762 1468 y Fn(K)f(>)25 b Fo(0)33 b Fh(such)g(that)1179 1637 y Fo(sup)1188 1725 y Fm(t)p Fg(2)1272 1708 y Fe(\026)1260 1725 y Fm(S)1331 1637 y Fd(j)p Fn(@)1409 1596 y Fm(n)1452 1605 y Fb(0)1404 1661 y Fm(t)1492 1637 y Fn(X)1567 1651 y Fm(m)1629 1660 y Fb(1)1668 1637 y Fo(\()p Fn(t)p Fo(\))p Fd(j)26 b(\024)f Fn(C)2014 1614 y Fo(~)1990 1637 y Fn(K)2074 1599 y Fm(n)2117 1608 y Fb(0)2151 1599 y Fe(+)p Fm(m)2268 1608 y Fb(1)2307 1637 y Fo(\()p Fn(K)7 b(n)2481 1651 y Fe(0)2520 1637 y Fo(\)!)p Fn(m)2660 1651 y Fe(1)2700 1637 y Fo(!)p Fn(;)75 1865 y Fh(for)33 b(al)5 b(l)33 b Fo(0)26 b Fd(\024)f Fn(m)597 1879 y Fe(1)661 1865 y Fd(\024)g Fn(S)h Fd(\000)20 b Fo(1)33 b Fh(and)g(al)5 b(l)33 b Fn(n)1369 1879 y Fe(0)1433 1865 y Fd(2)25 b Fl(N)7 b Fh(.)216 1978 y(Then)34 b(the)h(formal)g(series)1181 1955 y Fo(^)1155 1978 y Fn(X)8 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fh(,)35 b(solution)g(of)f Fo(\()p Fn(E)2045 1992 y Fm(P)10 b(V)2161 1978 y Fo(\))p Fh(,)34 b(is)g Fo(1)p Fn(=)p Fo(\()p Fn(K)29 b Fd(\000)21 b Fo(1\))p Fh(-summable)35 b(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)g Fn(t)75 2091 y Fh(in)e(the)g(dir)-5 b(e)g(ction)34 b Fn(d)25 b Fd(2)g Fl(R)s Fh(.)75 2372 y Fp(3)135 b(Analytic)38 b(con)l(tin)l(uation)h(of) f(solutions)h(of)f(linear)h(partial)g(di\013eren)l(tial)277 2522 y(equations)75 2725 y Fo(Let)31 b Fn(K)h Fd(\025)25 b Fo(1)31 b(b)s(e)e(an)i(in)m(teger.)41 b(W)-8 b(e)31 b(denote)g(b)m(y)f Fd(F)1745 2739 y Fm(K)1844 2725 y Fo(the)h(v)m(ector)h(space)f(of)f(formal)g(series)g(of)g(the)h(form)75 2947 y(\()p Fn(V)21 b Fo(\))1244 b Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1847 2860 y Ff(X)1852 3056 y Fm(j)t Fg(\025)p Fe(0)2048 2885 y Fn(t)2081 2852 y Fm(K)5 b(j)p 2004 2926 223 4 v 2004 3009 a Fo(\()p Fn(K)i(j)e Fo(\)!)2236 2947 y Fn(u)2288 2961 y Fm(j)2325 2947 y Fo(\()p Fn(z)t Fo(\))p Fn(;)75 3206 y Fo(where)29 b Fn(u)389 3220 y Fm(j)426 3206 y Fo(,)g Fn(j)i Fd(\025)25 b Fo(0,)31 b(are)f(holomorphic)d(functions)h(on)i(a)g(common)f(neigh)m(b)s(orho)s (o)s(d)f(\012)h(of)g(the)h(origin)e(in)g Fl(C)18 b Fo(.)46 b(It)75 3319 y(is)32 b(easy)i(to)g(see)f(that)h Fd(F)900 3333 y Fm(K)1002 3319 y Fo(is)e(stable)h(b)m(y)g(m)m(ultiplication)d(b) m(y)j Fn(t)2239 3286 y Fm(K)2340 3319 y Fo(and)f(b)m(y)h(deriv)-5 b(ation)32 b(with)g(resp)s(ect)h(to)g Fn(t)g Fo(of)75 3432 y(order)d Fn(K)7 b Fo(.)40 b(That)31 b(is,)e(if)h Fn(V)45 b Fd(2)25 b(F)1139 3446 y Fm(K)1208 3432 y Fo(,)30 b(then)g Fn(t)1503 3399 y Fm(K)1571 3432 y Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b Fd(2)f(F)2011 3446 y Fm(K)2110 3432 y Fo(and)30 b Fn(@)2340 3399 y Fm(K)2335 3454 y(t)2408 3432 y Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b Fd(2)f(F)2848 3446 y Fm(K)2917 3432 y Fo(.)75 3545 y(Consider)k(the)h(linear)f(application)g Fi(P)1405 3559 y Fm(K)1473 3545 y Fo(,)i(whic)m(h)e(is)g(a)i(bijection)e(from)h Fd(F)2607 3559 y Fm(K)2706 3545 y Fo(in)m(to)h Fd(F)2956 3559 y Fe(1)2995 3545 y Fo(,)g(de\014ned)e(b)m(y)1434 3765 y Fi(P)1505 3779 y Fm(K)1574 3765 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)2029 3679 y Ff(X)2033 3875 y Fm(j)t Fg(\025)p Fe(0)2186 3704 y Fn(t)2219 3671 y Fm(j)p 2186 3744 70 4 v 2187 3828 a Fn(j)5 b Fo(!)2265 3765 y Fn(u)2317 3779 y Fm(j)2353 3765 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 4024 y Fo(In)25 b(the)i(next)f(prop)s(osition,)f(w)m(e) h(giv)m(e)h(imp)s(ortan)m(t)e(comm)m(utations)i(form)m(ulas)e(for)g (the)i(applications)d Fi(P)3583 4038 y Fm(K)3678 4024 y Fo(with)75 4137 y(the)31 b(op)s(erations)e Fn(t)707 4104 y Fm(K)775 4137 y Fo(,)i Fn(@)884 4104 y Fm(K)879 4160 y(t)983 4137 y Fo(and)f Fn(a)p Fo(\()p Fn(z)t Fo(\))p Fn(@)1377 4104 y Fm(l)1372 4160 y(z)1443 4137 y Fo(that)h(will)d(b)s(e) i(useful)e(in)h(the)i(sequel.)75 4296 y Fi(Prop)s(osition)36 b(2)46 b(a\))67 b Fh(L)-5 b(et)33 b Fn(V)45 b Fd(2)25 b(F)1303 4310 y Fm(K)1372 4296 y Fh(,)32 b Fn(K)g Fd(\025)25 b Fo(1)p Fh(.)42 b(Then,)1164 4465 y Fi(P)1235 4479 y Fm(K)1304 4465 y Fo(\()p Fn(a)p Fo(\()p Fn(z)t Fo(\))p Fn(@)1556 4427 y Fm(l)1551 4487 y(z)1593 4465 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)e Fn(a)p Fo(\()p Fn(z)t Fo(\))p Fn(@)2230 4427 y Fm(l)2225 4487 y(z)2266 4465 y Fi(P)2337 4479 y Fm(K)2406 4465 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))p Fn(;)75 4634 y Fh(wher)-5 b(e)34 b Fn(a)p Fo(\()p Fn(z)t Fo(\))f Fh(is)g(an)g(holomorphic)j (function)d(on)g Fo(\012)p Fh(.)75 4747 y Fi(b\))f Fh(L)-5 b(et)33 b Fn(V)46 b Fd(2)25 b(F)613 4761 y Fm(K)681 4747 y Fh(,)33 b Fn(K)f Fd(\025)25 b Fo(1)p Fh(.)42 b(Then,)1318 4916 y Fi(P)1389 4930 y Fm(K)1458 4916 y Fo(\()p Fn(@)1546 4878 y Fm(K)1541 4939 y(t)1615 4916 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)e Fn(@)2083 4930 y Fm(t)2113 4916 y Fi(P)2184 4930 y Fm(K)2252 4916 y Fo(\()p Fn(V)c Fo(\()p Fn(t;)15 b(z)t Fo(\)\))p Fn(:)75 5085 y Fi(c\))38 b Fh(L)-5 b(et)39 b Fn(K)i Fd(\025)35 b Fo(1)p Fh(.)59 b(Then,)40 b(ther)-5 b(e)39 b(exist)f Fn(a)1488 5099 y Fm(q)1561 5085 y Fd(2)d Fl(C)18 b Fh(,)45 b Fo(0)36 b Fd(\024)f Fn(q)j Fd(\024)d Fn(K)c Fd(\000)24 b Fo(1)p Fh(,)39 b(which)g(dep)-5 b(end)40 b(on)e Fn(K)7 b Fh(,)39 b(such)g(that)g(the)75 5198 y(fol)5 b(lowing)34 b(holds.)43 b(L)-5 b(et)33 b Fn(V)46 b Fd(2)24 b(F)1136 5212 y Fm(K)1205 5198 y Fh(,)33 b Fn(K)e Fd(\025)25 b Fo(1)p Fh(.)1026 5467 y Fi(P)1097 5481 y Fm(K)1166 5467 y Fo(\()p Fn(t)1234 5429 y Fm(K)1302 5467 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)1722 5284 y Ff(0)1722 5448 y(@)1802 5353 y Fm(K)5 b Fg(\000)p Fe(1)1813 5380 y Ff(X)1817 5575 y Fm(q)r Fe(=0)1971 5467 y Fn(a)2019 5481 y Fm(q)2057 5467 y Fn(t)p Fo(\()p Fn(t@)2206 5481 y Fm(t)2236 5467 y Fo(\))2271 5429 y Fm(q)2310 5284 y Ff(1)2310 5448 y(A)2404 5467 y Fi(P)2475 5481 y Fm(K)2544 5467 y Fo(\()p Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))p Fn(;)75 5740 y Fh(wher)-5 b(e)34 b Fo(\()p Fn(t@)448 5754 y Fm(t)478 5740 y Fo(\))513 5707 y Fm(q)584 5740 y Fh(is)e(the)h Fn(q)s Fh(-th)g(c)-5 b(omp)g(osition)35 b(of)e(the)g(op)-5 b(er)g(ator)36 b Fn(t@)2214 5754 y Fm(t)2244 5740 y Fh(.)p eop %%Page: 4 4 4 3 bop 3809 100 a Fo(4)75 399 y Fi(Pro)s(of)31 b Fo(The)f(part)g Fi(a\))g Fo(follo)m(ws)g(directly)f(from)h(the)g(de\014nition)e(of)j Fi(P)2445 413 y Fm(K)2513 399 y Fo(.)41 b(F)-8 b(or)31 b(the)g(part)f Fi(b\))p Fo(,)g(let)1463 661 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1847 574 y Ff(X)1852 770 y Fm(j)t Fg(\025)p Fe(0)2048 599 y Fn(t)2081 566 y Fm(K)5 b(j)p 2004 640 223 4 v 2004 723 a Fo(\()p Fn(K)i(j)e Fo(\)!)2236 661 y Fn(u)2288 675 y Fm(j)2325 661 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 955 y Fo(Then)885 1108 y Fn(@)938 1070 y Fm(K)933 1130 y(t)1007 1108 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1392 1021 y Ff(X)1396 1217 y Fm(j)t Fg(\025)p Fe(0)1624 1046 y Fn(x)1676 1013 y Fm(K)5 b Fe(\()p Fm(j)t Fg(\000)p Fe(1\))p 1548 1087 450 4 v 1548 1170 a Fo(\()p Fn(K)i Fo(\()p Fn(j)26 b Fd(\000)20 b Fo(1\)\)!)2008 1108 y Fn(u)2060 1122 y Fm(j)2096 1108 y Fo(\()p Fn(z)t Fo(\))26 b(=)2334 1021 y Ff(X)2339 1217 y Fm(j)t Fg(\025)p Fe(0)2535 1046 y Fn(t)2568 1013 y Fm(K)5 b(j)p 2491 1087 223 4 v 2491 1170 a Fo(\()p Fn(K)i(j)e Fo(\)!)2723 1108 y Fn(u)2775 1122 y Fm(j)t Fe(+1)2902 1108 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 1365 y Fo(On)30 b(the)g(other)h(hand,)e(let)1540 1529 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1923 1443 y Ff(X)1928 1639 y Fm(j)t Fg(\025)p Fe(0)2080 1468 y Fn(t)2113 1435 y Fm(j)p 2080 1508 70 4 v 2081 1591 a Fn(j)5 b Fo(!)2159 1529 y Fn(u)2211 1543 y Fm(j)2247 1529 y Fo(\()p Fn(z)t Fo(\))p Fn(;)75 1786 y Fo(and)30 b(as)g(b)s(efore,)1456 1950 y Fn(@)1504 1964 y Fm(t)1534 1950 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1917 1864 y Ff(X)1921 2060 y Fm(j)t Fg(\025)p Fe(0)2073 1889 y Fn(t)2106 1856 y Fm(j)p 2073 1929 V 2074 2013 a Fn(j)5 b Fo(!)2153 1950 y Fn(u)2205 1964 y Fm(j)t Fe(+1)2332 1950 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 2212 y Fo(P)m(art)31 b Fi(c\))p Fo(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(that)255 2479 y Fn(t)288 2441 y Fm(K)356 2479 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)741 2393 y Ff(X)745 2589 y Fm(j)t Fg(\025)p Fe(0)897 2418 y Fn(t)930 2385 y Fm(K)5 b Fe(\()p Fm(j)t Fe(+1\))p 897 2458 279 4 v 925 2541 a Fo(\()p Fn(K)i(j)e Fo(\)!)1186 2479 y Fn(u)1238 2493 y Fm(j)1274 2479 y Fo(\()p Fn(z)t Fo(\))26 b(=)1512 2393 y Ff(X)1517 2589 y Fm(j)t Fg(\025)p Fe(0)1754 2418 y Fn(t)1787 2385 y Fm(K)5 b Fe(\()p Fm(j)t Fe(+1\))p 1669 2458 450 4 v 1669 2541 a Fo(\()p Fn(K)i Fo(\()p Fn(j)26 b Fo(+)20 b(1\)\)!)2128 2479 y Fn(M)2216 2493 y Fm(K)2285 2479 y Fo(\()p Fn(j)5 b Fo(\))p Fn(u)2449 2493 y Fm(j)2487 2479 y Fo(\()p Fn(z)t Fo(\))26 b(=)2725 2393 y Ff(X)2729 2589 y Fm(j)t Fg(\025)p Fe(0)2926 2418 y Fn(t)2959 2385 y Fm(K)5 b(j)p 2881 2458 223 4 v 2881 2541 a Fo(\()p Fn(K)i(j)e Fo(\)!)3114 2479 y Fn(P)3172 2493 y Fm(K)3240 2479 y Fo(\()p Fn(j)g Fo(\))p Fn(u)3404 2493 y Fm(j)t Fg(\000)p Fe(1)3533 2479 y Fo(\()p Fn(z)t Fo(\))p Fn(;)75 2778 y Fo(where)25 b Fn(P)391 2792 y Fm(K)460 2778 y Fo(\()p Fn(j)5 b Fo(\))27 b(=)e(\()p Fn(K)7 b(j)e Fo(\)\()p Fn(K)i(j)18 b Fd(\000)11 b Fo(1\))k Fd(\001)g(\001)g(\001)i Fo(\()p Fn(K)7 b(j)17 b Fd(\000)11 b Fn(K)18 b Fo(+)11 b(1\))26 b(is)f(a)i(p)s(olynomial)c(of)j(order)f Fn(K)33 b Fo(in)25 b(the)h(v)-5 b(ariable)24 b Fn(j)5 b Fo(.)40 b(Note)75 2891 y(that)31 b Fn(P)330 2905 y Fm(K)399 2891 y Fo(\(0\))26 b(=)f(0.)216 3054 y(On)30 b(the)g(other)h(hand,)e(w)m(e)i(ha)m(v)m(e)h(that)1092 3310 y Fn(t@)1173 3324 y Fm(t)1202 3310 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1586 3223 y Ff(X)1590 3419 y Fm(j)t Fg(\025)p Fe(0)1742 3248 y Fn(t)1775 3215 y Fm(j)t Fe(+1)p 1742 3289 160 4 v 1788 3372 a Fn(j)5 b Fo(!)1912 3310 y Fn(u)1964 3324 y Fm(j)t Fe(+1)2090 3310 y Fo(\()p Fn(z)t Fo(\))27 b(=)2329 3223 y Ff(X)2333 3419 y Fm(j)t Fg(\025)p Fe(0)2485 3248 y Fn(t)2518 3215 y Fm(j)p 2485 3289 70 4 v 2486 3372 a Fn(j)5 b Fo(!)2564 3310 y Fn(j)g(u)2658 3324 y Fm(j)2696 3310 y Fo(\()p Fn(z)t Fo(\))p Fn(;)75 3604 y Fo(so)31 b(that,)g(for)f(all)f Fn(q)f Fd(\025)d Fo(0,)1390 3768 y(\()p Fn(t@)1506 3782 y Fm(t)1536 3768 y Fo(\))1571 3731 y Fm(q)1609 3768 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1993 3682 y Ff(X)1997 3878 y Fm(j)t Fg(\025)p Fe(0)2149 3707 y Fn(t)2182 3674 y Fm(j)p 2149 3748 V 2150 3831 a Fn(j)5 b Fo(!)2228 3768 y Fn(j)2270 3731 y Fm(q)2309 3768 y Fn(u)2361 3782 y Fm(j)2398 3768 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 4025 y Fo(Moreo)m(v)m(er,)33 b(as)d(b)s(efore)1457 4190 y Fn(tU)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1873 4103 y Ff(X)1878 4299 y Fm(j)t Fg(\025)p Fe(0)2030 4128 y Fn(t)2063 4095 y Fm(j)p 2030 4169 V 2031 4252 a Fn(j)5 b Fo(!)2109 4190 y Fn(j)g(u)2203 4204 y Fm(j)t Fg(\000)p Fe(1)2330 4190 y Fo(\()p Fn(z)t Fo(\))p Fn(:)75 4447 y Fo(F)-8 b(rom)31 b(the)f(latter)h(form)m(ulas,)f(w)m(e)h(deduce)f (that)h(for)f(all)f Fn(q)f Fd(\025)d Fo(0,)1265 4702 y Fn(t)p Fo(\()p Fn(t@)1414 4716 y Fm(t)1444 4702 y Fo(\))1479 4665 y Fm(q)1518 4702 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1901 4616 y Ff(X)1906 4812 y Fm(j)t Fg(\025)p Fe(0)2058 4641 y Fn(t)2091 4608 y Fm(j)p 2058 4682 V 2059 4765 a Fn(j)5 b Fo(!)2137 4702 y Fn(\026)2192 4716 y Fm(q)2230 4702 y Fo(\()p Fn(j)g Fo(\))p Fn(u)2394 4716 y Fm(j)t Fg(\000)p Fe(1)2522 4702 y Fo(\()p Fn(z)t Fo(\))p Fn(;)75 5002 y Fo(where)30 b Fn(\026)393 5016 y Fm(q)431 5002 y Fo(\()p Fn(j)5 b Fo(\))26 b(=)f Fn(j)5 b Fo(\()p Fn(j)27 b Fd(\000)20 b Fo(1\))977 4969 y Fm(q)1046 5002 y Fo(is)30 b(a)g(p)s(olynomial)e(of)j(order)e Fn(q)23 b Fo(+)d(1)31 b(in)e(the)i(v)-5 b(ariable)29 b Fn(j)5 b Fo(.)41 b(Note)32 b(that)f Fn(\026)3439 5016 y Fm(q)3477 5002 y Fo(\(0\))26 b(=)f(0.)216 5165 y(No)m(w,)k(the)f(p)s(olynomial)d Fn(P)1125 5179 y Fm(K)1221 5165 y Fo(can)j(b)s(e)f(written)g(as)h(a)g (linear)e(com)m(bination)h(with)f(co)s(e\016cien)m(ts)j(in)d Fl(C)51 b Fo(of)28 b(the)75 5277 y(p)s(olynomials)g Fn(\026)636 5291 y Fm(q)673 5277 y Fo(,)j(0)26 b Fd(\024)f Fn(q)j Fd(\024)d Fn(K)i Fd(\000)19 b Fo(1,)1541 5480 y Fn(P)1599 5494 y Fm(K)1667 5480 y Fo(\()p Fn(j)5 b Fo(\))27 b(=)1902 5367 y Fm(K)5 b Fg(\000)p Fe(1)1913 5394 y Ff(X)1917 5588 y Fm(q)r Fe(=0)2071 5480 y Fn(a)2119 5494 y Fm(q)2158 5480 y Fn(\026)2213 5494 y Fm(q)2250 5480 y Fo(\()p Fn(j)g Fo(\))p Fn(;)p eop %%Page: 5 5 5 4 bop 3809 100 a Fo(5)75 399 y(where)30 b Fn(a)386 413 y Fm(q)449 399 y Fd(2)25 b Fl(C)18 b Fo(,)36 b(for)30 b(all)g Fn(j)h Fd(\025)25 b Fo(0.)41 b(F)-8 b(rom)31 b(this)e(w)m(e)i(deduce)f(that)999 522 y Ff(0)999 686 y(@)1079 591 y Fm(K)5 b Fg(\000)p Fe(1)1090 619 y Ff(X)1094 813 y Fm(q)r Fe(=0)1248 705 y Fn(a)1296 719 y Fm(q)1334 705 y Fn(t)p Fo(\()p Fn(t@)1483 719 y Fm(t)1513 705 y Fo(\))1548 667 y Fm(q)1586 522 y Ff(1)1586 686 y(A)1681 705 y Fn(U)10 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)2065 619 y Ff(X)2069 815 y Fm(j)t Fg(\025)p Fe(0)2256 643 y Fn(t)2289 610 y Fm(j)p 2221 684 139 4 v 2221 767 a Fo(\()p Fn(j)5 b Fo(\)!)2370 705 y Fn(P)2428 719 y Fm(K)2496 705 y Fo(\()p Fn(j)g Fo(\))p Fn(u)2660 719 y Fm(j)t Fg(\000)p Fe(1)2788 705 y Fo(\()p Fn(z)t Fo(\))p Fn(:)3756 980 y Fa(2)75 1141 y Fo(In)37 b(the)g(next)h(prop)s(osition,)f(w)m(e)h (study)e(the)i(linear)e(partial)g(di\013eren)m(tial)g(equations)h (satis\014ed)f(b)m(y)i Fi(P)3642 1155 y Fm(K)3711 1141 y Fo(\()p Fn(V)20 b Fo(\))75 1253 y(when)29 b Fn(V)51 b Fo(satis\014es)30 b(itself)f(a)i(linear)d(partial)i(di\013eren)m (tial)e(equation.)75 1436 y Fi(Prop)s(osition)36 b(3)46 b Fh(Assume)35 b(that)h(a)g(formal)h(series)e Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))31 b Fd(2)f(F)2350 1450 y Fm(K)2454 1436 y Fh(satis\014es)36 b(a)f(line)-5 b(ar)37 b(p)-5 b(artial)37 b(di\013er)-5 b(ential)75 1549 y(e)g(quation)75 1756 y Fo(\()p Fn(E)177 1770 y Fm(V)238 1756 y Fo(\))751 b Fn(@)1077 1718 y Fm(S)1072 1778 y(z)1128 1756 y Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1588 1669 y Ff(X)1513 1871 y Fe(\()p Fm(l)1561 1880 y Fb(0)1596 1871 y Fm(;l)1637 1880 y Fb(1)1671 1871 y Fe(\))p Fg(2S)1809 1756 y Fn(B)1878 1774 y Fe(\()p Fm(l)1926 1783 y Fb(0)1961 1774 y Fm(;l)2002 1783 y Fb(1)2036 1774 y Fe(\))2068 1756 y Fo(\()p Fn(t)2136 1718 y Fm(K)2204 1756 y Fn(;)15 b(z)t Fo(\))p Fn(@)2378 1715 y Fm(K)5 b(l)2464 1724 y Fb(0)2373 1780 y Fm(t)2503 1756 y Fn(@)2556 1718 y Fm(l)2577 1727 y Fb(1)2551 1778 y Fm(z)2617 1756 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(;)75 2068 y Fh(wher)-5 b(e)38 b Fd(S)44 b Fh(is)37 b(a)h(\014nite)f(subset)g(of)g Fl(N)1305 2035 y Fe(2)1388 2068 y Fh(and)h Fn(B)1638 2087 y Fe(\()p Fm(l)1686 2096 y Fb(0)1721 2087 y Fm(;l)1762 2096 y Fb(1)1796 2087 y Fe(\))1828 2068 y Fo(\()p Fn(t)1896 2035 y Fm(K)1964 2068 y Fn(;)15 b(z)t Fo(\))38 b Fh(ar)-5 b(e)38 b(p)-5 b(olynomials)41 b(with)d(r)-5 b(esp)g(e)g(ct)38 b(to)g Fn(t)3431 2035 y Fm(K)3536 2068 y Fh(of)g(or)-5 b(der)75 2181 y Fn(d)122 2200 y Fe(\()p Fm(l)170 2209 y Fb(0)205 2200 y Fm(;l)246 2209 y Fb(1)281 2200 y Fe(\))345 2181 y Fh(with)33 b(holomorphic)j(c)-5 b(o)g(e\016cients)34 b(in)e Fo(\012)p Fh(.)216 2294 y(Then,)d(the)g(formal)g(series)f Fi(P)1231 2308 y Fm(K)1300 2294 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))30 b Fh(satis\014es)f(a)f(line)-5 b(ar)29 b(p)-5 b(artial)30 b(di\013er)-5 b(ential)29 b(e)-5 b(quation)29 b(of)f(the)h(form)75 2506 y Fo(\()p Fn(E)177 2520 y Fm(P)10 b(V)293 2506 y Fo(\))491 b Fn(@)872 2468 y Fm(S)867 2528 y(z)923 2506 y Fi(P)994 2520 y Fm(K)1062 2506 y Fo(\()p Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)1623 2419 y Ff(X)1518 2621 y Fe(\()p Fm(k)1582 2630 y Fb(0)1617 2621 y Fm(;k)1674 2630 y Fb(1)1708 2621 y Fe(\))p Fg(2S)1825 2630 y Fb(1)1874 2506 y Fn(A)1942 2524 y Fe(\()p Fm(k)2006 2533 y Fb(0)2041 2524 y Fm(;k)2098 2533 y Fb(1)2132 2524 y Fe(\))2164 2506 y Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(@)2406 2465 y Fm(k)2443 2474 y Fb(0)2401 2530 y Fm(t)2483 2506 y Fn(@)2536 2468 y Fm(k)2573 2477 y Fb(1)2531 2528 y Fm(z)2612 2506 y Fi(P)2683 2520 y Fm(K)2752 2506 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))p Fn(;)75 2816 y Fh(wher)-5 b(e)38 b Fd(S)391 2830 y Fe(1)467 2816 y Fh(is)f(a)g(\014nite)g(subset)g(of)g Fl(N)1335 2783 y Fe(2)1418 2816 y Fh(and)g Fn(A)1666 2834 y Fe(\()p Fm(k)1730 2843 y Fb(0)1765 2834 y Fm(;k)1822 2843 y Fb(1)1856 2834 y Fe(\))1888 2816 y Fo(\()p Fn(t;)15 b(z)t Fo(\))38 b Fh(ar)-5 b(e)38 b(p)-5 b(olynomials)40 b(with)d(r)-5 b(esp)g(e)g(ct)39 b(to)e Fn(t)g Fh(with)g(holo-)75 2929 y(morphic)29 b(c)-5 b(o)g(e\016cients)29 b(in)e Fo(\012)p Fh(.)40 b(Mor)-5 b(e)g(over,)30 b(the)e(highest)g(p)-5 b(ower)30 b Fn(M)2337 2943 y Fe(0)2376 2929 y Fo(\()p Fn(l)2438 2943 y Fe(1)2478 2929 y Fo(\))e Fh(of)g Fn(@)2691 2943 y Fm(t)2748 2929 y Fh(which)h(app)-5 b(e)g(ars)31 b(in)c(monomials)75 3042 y(of)33 b(the)g(form)g Fn(@)605 3001 y Fm(k)642 3010 y Fb(0)600 3066 y Fm(t)681 3042 y Fn(@)734 3009 y Fm(l)755 3018 y Fb(1)729 3064 y Fm(z)827 3042 y Fh(for)g(\014xe)-5 b(d)34 b Fn(l)1209 3056 y Fe(1)1281 3042 y Fh(in)e(the)h(right)g(handside)i(of)e Fo(\()p Fn(E)2346 3056 y Fm(P)10 b(V)2462 3042 y Fo(\))32 b Fh(is)h(of)g(the)g (form)1210 3240 y Fn(M)1298 3254 y Fe(0)1338 3240 y Fo(\()p Fn(l)1400 3254 y Fe(1)1440 3240 y Fo(\))26 b(=)80 b(max)1597 3305 y Fe(\()p Fm(l)1645 3314 y Fb(0)1680 3305 y Fm(;l)1721 3314 y Fb(1)1755 3305 y Fe(\))p Fg(2S)1877 3240 y Fd(f)p Fn(d)1969 3258 y Fe(\()p Fm(l)2017 3267 y Fb(0)2053 3258 y Fm(;l)2094 3267 y Fb(1)2128 3258 y Fe(\))2160 3240 y Fo(\()p Fn(K)27 b Fd(\000)20 b Fo(1\))h(+)f Fn(l)2609 3254 y Fe(0)2648 3240 y Fd(g)p Fn(:)75 3540 y Fi(Pro)s(of)26 b Fo(Apply)d(the)i(linear)e(op)s(erator)i Fi(P)1451 3554 y Fm(K)1545 3540 y Fo(the)g(equation)g(\()p Fn(E)2164 3554 y Fm(V)2225 3540 y Fo(\).)39 b(The)25 b(result)e(follo)m(ws)h (from)h(the)g(prop)s(osition)75 3653 y(2.)3611 b Fa(2)75 3814 y Fo(The)24 b(next)h(result)f(giv)m(es)i(su\016cien)m(t)e (conditions)f(for)i(the)g(existence)g(of)g(analytic)g(solutions)e Fn(V)e Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(of)f(partial)75 3926 y(di\013eren)m(tial)31 b(equations)h(of)g(Cauc)m(h)m(y-Ko)m(w)m (alevsk)-5 b(a)m(y)m(a)36 b(t)m(yp)s(e)d(in)e(a)h(neigh)m(b)s(orho)s(o) s(d)e(of)j(the)g(origin)d(in)h Fl(C)3610 3893 y Fe(2)3688 3926 y Fo(that)75 4039 y(can)d(b)s(e)g(analytically)f(con)m(tin)m(ued)g (with)g(resp)s(ect)h(to)h(the)f(v)-5 b(ariable)27 b Fn(t)h Fo(in)f(sectors)i(of)f(in\014nite)e(radius.)38 b(Gro)m(wth)75 4152 y(estimates)31 b(for)f(these)h(solutions)e(are)i(also)f(obtained.) 75 4359 y Fi(Theorem)k(1)46 b Fh(L)-5 b(et)28 b Fn(K)k(>)25 b Fo(1)k Fh(an)g(inte)-5 b(ger.)40 b(Consider)30 b(a)f(formal)h(series) e Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b Fd(2)e(F)2945 4373 y Fm(K)3041 4359 y Fh(solution)30 b(of)f(the)f(line)-5 b(ar)75 4472 y(p)g(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation,)75 4678 y Fo(\()p Fn(E)177 4692 y Fm(V)238 4678 y Fo(\))751 b Fn(@)1077 4641 y Fm(S)1072 4701 y(z)1128 4678 y Fn(V)21 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)1588 4592 y Ff(X)1513 4793 y Fe(\()p Fm(l)1561 4802 y Fb(0)1596 4793 y Fm(;l)1637 4802 y Fb(1)1671 4793 y Fe(\))p Fg(2S)1809 4678 y Fn(B)1878 4697 y Fe(\()p Fm(l)1926 4706 y Fb(0)1961 4697 y Fm(;l)2002 4706 y Fb(1)2036 4697 y Fe(\))2068 4678 y Fo(\()p Fn(t)2136 4641 y Fm(K)2204 4678 y Fn(;)15 b(z)t Fo(\))p Fn(@)2378 4638 y Fm(K)5 b(l)2464 4647 y Fb(0)2373 4702 y Fm(t)2503 4678 y Fn(@)2556 4641 y Fm(l)2577 4650 y Fb(1)2551 4701 y Fm(z)2617 4678 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(;)75 4991 y Fh(wher)-5 b(e)35 b Fn(B)402 5009 y Fe(\()p Fm(l)450 5018 y Fb(0)486 5009 y Fm(;l)527 5018 y Fb(1)561 5009 y Fe(\))592 4991 y Fo(\()p Fn(t)660 4958 y Fm(K)729 4991 y Fn(;)15 b(z)t Fo(\))35 b Fh(ar)-5 b(e)36 b(p)-5 b(olynomials)37 b(with)e(r)-5 b(esp)g(e)g(ct)36 b(to)f Fn(t)2179 4958 y Fm(K)2282 4991 y Fh(of)f(or)-5 b(der)37 b Fn(d)2679 5009 y Fe(\()p Fm(l)2727 5018 y Fb(0)2762 5009 y Fm(;l)2803 5018 y Fb(1)2837 5009 y Fe(\))2903 4991 y Fh(with)e(holomorphic)j(c)-5 b(o)g(e\016-)75 5104 y(cients)33 b(in)f Fo(\012)g Fh(and)i Fd(S)39 b Fh(is)33 b(a)g(\014nite)f(subset)h(of)g Fl(N)1665 5071 y Fe(2)1743 5104 y Fh(of)g(the)g(form)806 5302 y Fd(S)f Fo(=)25 b Fd(f)p Fo(\()p Fn(l)1096 5316 y Fe(0)1136 5302 y Fn(;)15 b(l)1203 5316 y Fe(1)1243 5302 y Fo(\))26 b Fd(2)f Fl(N)1449 5264 y Fe(2)1520 5302 y Fo(:)h Fn(d)1618 5317 y Fm(l)1639 5326 y Fb(0)1674 5317 y Fm(;l)1715 5326 y Fb(1)1753 5302 y Fn(K)7 b Fo(\()p Fn(K)27 b Fd(\000)20 b Fo(1\))h(+)f Fn(K)7 b(l)2370 5316 y Fe(0)2429 5302 y Fo(+)20 b Fn(l)2547 5316 y Fe(1)2612 5302 y Fd(\024)25 b Fn(S;)15 b(l)2831 5316 y Fe(1)2896 5302 y Fn(<)25 b(S)5 b Fd(g)p Fn(:)75 5500 y Fh(L)-5 b(et)1214 5656 y Fi(P)1285 5670 y Fm(K)1354 5656 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)1835 5633 y(^)1809 5656 y Fn(X)8 b Fo(\()p Fn(t;)15 b(z)t Fo(\))26 b(=)2204 5570 y Ff(X)2203 5766 y Fm(n)p Fg(\025)p Fe(0)2387 5572 y Fo(^)2361 5595 y Fn(X)2436 5609 y Fm(n)2483 5595 y Fo(\()p Fn(t)p Fo(\))p 2361 5635 226 4 v 2434 5718 a Fn(n)p Fo(!)2597 5656 y Fn(z)2643 5619 y Fm(n)2690 5656 y Fn(:)p eop %%Page: 6 6 6 5 bop 3809 100 a Fo(6)75 399 y Fh(We)27 b(assume)h(that)g(the)g (formal)h(series)1435 376 y Fo(^)1409 399 y Fn(X)1484 413 y Fm(\026)1531 399 y Fo(\()p Fn(t)p Fo(\))p Fh(,)g(for)e Fo(0)f Fd(\024)f Fn(\026)g Fd(\024)g Fn(S)14 b Fd(\000)9 b Fo(1)p Fh(,)27 b(ar)-5 b(e)29 b Fo(1)p Fn(=)p Fo(\()p Fn(K)15 b Fd(\000)9 b Fo(1\))p Fh(-summable)29 b(with)f(r)-5 b(esp)g(e)g(ct)75 511 y(to)33 b Fn(t)g Fh(in)f(a)h(dir)-5 b(e)g(ction)34 b Fn(d)26 b Fd(2)f Fl(R)r Fh(.)216 624 y(Mor)-5 b(e)g(over,)51 b(we)46 b(assume)h(that)h(their)f(sums)f Fn(X)1890 638 y Fm(\026)1937 624 y Fo(\()p Fn(t)p Fo(\))h Fh(ar)-5 b(e)47 b(holomorphic)i(in)e(a)f(se)-5 b(ctorial)48 b(r)-5 b(e)g(gion)47 b Fn(G)k Fd(\032)75 737 y Fn(D)s Fo(\(0)p Fn(;)15 b(R)q Fo(\()p Fn(\032)p Fo(\)\))p Fh(,)31 b(indep)-5 b(endent)31 b(of)f Fn(\026)p Fh(,)f(which)h(has)h(bise)-5 b(cting)29 b(dir)-5 b(e)g(ction)30 b Fn(d)g Fh(and)g(op)-5 b(ening)30 b(gr)-5 b(e)g(ater)31 b(than)f Fo(\()p Fn(K)20 b Fd(\000)13 b Fo(1\))p Fn(\031)75 850 y Fh(and)34 b(satisfy)f(the)g (fol)5 b(lowing)34 b(estimates.)75 963 y(F)-7 b(or)34 b(al)5 b(l)33 b(close)-5 b(d)34 b(se)-5 b(ctor)919 940 y Fo(\026)903 963 y Fn(S)31 b Fd(\032)25 b Fn(G)p Fh(,)32 b(ther)-5 b(e)34 b(exist)f Fn(C)q(;)1787 940 y Fo(~)1762 963 y Fn(K)f(>)25 b Fo(0)33 b Fh(such)g(that)1179 1167 y Fo(sup)1188 1255 y Fm(t)p Fg(2)1272 1238 y Fe(\026)1260 1255 y Fm(S)1331 1167 y Fd(j)p Fn(@)1409 1127 y Fm(n)1452 1136 y Fb(0)1404 1191 y Fm(t)1492 1167 y Fn(X)1567 1181 y Fm(m)1629 1190 y Fb(1)1668 1167 y Fo(\()p Fn(t)p Fo(\))p Fd(j)26 b(\024)f Fn(C)2014 1144 y Fo(~)1990 1167 y Fn(K)2074 1130 y Fm(n)2117 1139 y Fb(0)2151 1130 y Fe(+)p Fm(m)2268 1139 y Fb(1)2307 1167 y Fo(\()p Fn(K)7 b(n)2481 1181 y Fe(0)2520 1167 y Fo(\)!)p Fn(m)2660 1181 y Fe(1)2700 1167 y Fo(!)p Fn(;)75 1431 y Fh(for)33 b(al)5 b(l)33 b Fo(0)26 b Fd(\024)f Fn(m)597 1445 y Fe(1)661 1431 y Fd(\024)g Fn(S)h Fd(\000)20 b Fo(1)33 b Fh(and)g(al)5 b(l)33 b Fn(n)1369 1445 y Fe(0)1433 1431 y Fd(2)25 b Fl(N)7 b Fh(.)216 1544 y(Then,)37 b(the)g(formal)g(solution)g Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))38 b Fh(of)e Fo(\()p Fn(E)1791 1558 y Fm(V)1852 1544 y Fo(\))g Fh(c)-5 b(onver)g(ges)37 b(in)f(a)g(neighb)-5 b(orho)g(o)g(d)40 b(of)c(the)g(origin)h(in)f Fl(C)3781 1511 y Fe(2)3827 1544 y Fh(.)75 1657 y(Mor)-5 b(e)g(over,)39 b Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))38 b Fh(c)-5 b(an)37 b(b)-5 b(e)37 b(analytic)-5 b(al)5 b(ly)39 b(c)-5 b(ontinue)g(d)38 b(with)g(r)-5 b(esp)g(e)g(ct)38 b(to)g Fn(t)e Fh(in)h(a)g(se)-5 b(ctor)38 b Fn(S)3201 1675 y Fm(d=K)q(;\016)3423 1657 y Fo(=)32 b Fd(f)p Fn(t)h Fd(2)g Fl(C)56 b Fo(:)75 1770 y Fd(j)p Fn(d=K)28 b Fd(\000)20 b Fo(arg)r(\()p Fn(t)p Fo(\))p Fd(j)26 b Fn(<)f(\016)s Fd(g)p Fh(.)43 b(Mor)-5 b(e)g(over,)33 b(ther)-5 b(e)34 b(exist)f Fn(\016)1831 1784 y Fe(1)1896 1770 y Fn(<)25 b(\016)s Fh(,)33 b Fn(C)f(>)25 b Fo(0)p Fh(,)32 b(and)i Fn(M)h(>)25 b Fo(0)33 b Fh(such)g(that)1440 2031 y Fo(sup)1364 2119 y Fm(z)s Fg(2)1463 2102 y Fe(\026)1447 2119 y Fm(D)r Fe(\(0)p Fm(;\032)p Fe(\))1668 2031 y Fd(j)p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fd(j)27 b(\024)e Fn(C)7 b(e)2217 1994 y Fm(M)g Fg(j)p Fm(t)p Fg(j)2429 1926 y Fc(K)p 2366 1934 181 3 v 2366 1974 a Fb(\()p Fc(K)s Fk(\000)p Fb(1\))75 2308 y Fh(for)33 b(al)5 b(l)33 b Fn(t)25 b Fd(2)g Fn(S)550 2326 y Fm(d=K)q(;\016)733 2335 y Fb(1)771 2308 y Fh(.)75 2587 y Fi(Pro)s(of)31 b Fo(F)-8 b(rom)31 b(the)f(prop)s(osition)e(3,)j(w)m(e)g(deduce)f(that)h(the)f(formal)f (series)h Fi(P)2723 2601 y Fm(K)2792 2587 y Fo(\()p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)3273 2564 y(^)3247 2587 y Fn(X)8 b Fo(\()p Fn(t;)15 b(z)t Fo(\))31 b(satis\014es)75 2699 y(a)g(partial)e(di\013eren)m(tial)g(equation)h(of)g(the)h(form)75 2912 y(\()p Fn(E)177 2926 y Fm(P)10 b(V)293 2912 y Fo(\))692 b Fn(@)1073 2875 y Fm(S)1068 2935 y(z)1150 2889 y Fo(^)1124 2912 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)1623 2826 y Ff(X)1518 3027 y Fe(\()p Fm(k)1582 3036 y Fb(0)1617 3027 y Fm(;k)1674 3036 y Fb(1)1708 3027 y Fe(\))p Fg(2S)1825 3036 y Fb(1)1874 2912 y Fn(A)1942 2931 y Fe(\()p Fm(k)2006 2940 y Fb(0)2041 2931 y Fm(;k)2098 2940 y Fb(1)2132 2931 y Fe(\))2164 2912 y Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(@)2406 2872 y Fm(k)2443 2881 y Fb(0)2401 2936 y Fm(t)2483 2912 y Fn(@)2536 2875 y Fm(k)2573 2884 y Fb(1)2531 2935 y Fm(z)2638 2889 y Fo(^)2612 2912 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fn(;)75 3228 y Fo(where)28 b Fn(A)404 3247 y Fe(\()p Fm(k)468 3256 y Fb(0)503 3247 y Fm(;k)560 3256 y Fb(1)594 3247 y Fe(\))626 3228 y Fo(\()p Fn(t;)15 b(z)t Fo(\))29 b(are)g(holomorphic)e(functions)g(on)i(a)g(neigh)m(b)s(orho)s (o)s(d)d(of)j(the)f(origin)f(in)h Fl(C)3347 3195 y Fe(2)3421 3228 y Fo(and)g Fd(S)3651 3242 y Fe(1)3719 3228 y Fo(is)g(a)75 3341 y(\014nite)h(subset)h(of)h(the)f(form)1083 3545 y Fd(S)1138 3559 y Fe(1)1203 3545 y Fo(=)25 b Fd(f)p Fo(\()p Fn(k)1426 3559 y Fe(0)1466 3545 y Fn(;)15 b(k)1553 3559 y Fe(1)1594 3545 y Fo(\))25 b Fd(2)g Fl(N)1800 3508 y Fe(2)1870 3545 y Fo(:)h Fn(k)1968 3559 y Fe(0)2008 3545 y Fn(K)h Fo(+)19 b Fn(k)2249 3559 y Fe(1)2314 3545 y Fd(\024)25 b Fn(S;)15 b(k)2553 3559 y Fe(1)2619 3545 y Fn(<)25 b(S)5 b Fd(g)p Fn(:)75 3750 y Fo(F)-8 b(rom)39 b(the)f(assumptions)f(on)h(the)g(formal)g(series)1881 3727 y(^)1854 3750 y Fn(X)8 b Fo(\()p Fn(t;)15 b(z)t Fo(\),)42 b(w)m(e)c(then)g(deduce,)i(that)f(the)g(assumptions)d(of)75 3863 y(Prop)s(osition)25 b(1)j(are)f(ful\014lled.)36 b(So)27 b(that)h(the)g(formal)e(series)2162 3840 y(^)2136 3863 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\))28 b(is)f(1)p Fn(=)p Fo(\()p Fn(K)21 b Fd(\000)14 b Fo(1\)-summable)27 b(with)e(resp)s(ect)75 3975 y(to)31 b Fn(t)f Fo(in)f(the)i(direction)e Fn(d)p Fo(.)41 b(As)30 b(a)h(result,)f(the)g(Borel)h(transform)e(of)i (order)f(1)p Fn(=)p Fo(\()p Fn(K)e Fd(\000)20 b Fo(1\))31 b(of)3211 3952 y(^)3185 3975 y Fn(X)8 b Fo(,)1311 4249 y Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\))26 b(=)1751 4135 y Fg(1)1721 4163 y Ff(X)1725 4358 y Fm(j)t Fe(=0)2081 4187 y Fn(X)2156 4201 y Fm(j)2193 4187 y Fo(\()p Fn(z)t Fo(\))p Fn(t)2342 4154 y Fm(j)p 1878 4228 706 4 v 1878 4311 a Fn(j)5 b Fo(!\000\(1)21 b(+)f(\()p Fn(K)27 b Fd(\000)20 b Fo(1\))p Fn(j)5 b Fo(\))2593 4249 y Fn(;)75 4556 y Fo(con)m(v)m(erges)37 b(in)d(a)h(neigh)m(b)s(orho)s(o)s(d)e(of)i(the)g (origin)e(in)h Fl(C)1956 4523 y Fe(2)2036 4556 y Fo(and)g(can)i(b)s(e)e (analytically)g(con)m(tin)m(ued)g(in)g(a)h(sector)75 4669 y Fn(S)131 4684 y Fm(d;\016)225 4669 y Fo(,)30 b(with)f(exp)s (onen)m(tial)h(gro)m(wth)g(estimates)h(of)g(the)f(form)1451 4924 y(sup)1375 5012 y Fm(z)s Fg(2)1474 4995 y Fe(\026)1458 5012 y Fm(D)r Fe(\(0)p Fm(;\032)p Fe(\))1679 4924 y Fd(j)p Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\))p Fd(j)27 b(\024)e Fn(C)7 b(e)2254 4887 y Fm(M)g Fg(j)p Fm(t)p Fg(j)2454 4828 y Fb(1)p 2403 4836 133 3 v 2403 4873 a Fc(K)s Fk(\000)p Fb(1)75 5201 y Fo(for)30 b(all)f Fn(t)c Fd(2)g Fn(S)540 5216 y Fm(d;\016)627 5225 y Fb(1)666 5201 y Fo(,)31 b Fn(\016)762 5215 y Fe(1)827 5201 y Fn(<)25 b(\016)s Fo(.)216 5313 y(F)-8 b(rom)31 b(the)g(general)f(theory)h(of)f(momen)m(t)h (summabilit)m(y)d(\(see)j([1]\),)h(w)m(e)f(get)g(that)g(the)g(formal)f (series)1478 5551 y(~)1451 5574 y Fn(W)13 b Fo(\()p Fn(t;)i(z)t Fo(\))26 b(=)1861 5488 y Ff(X)1865 5684 y Fm(j)t Fg(\025)p Fe(0)2094 5513 y Fn(t)2127 5480 y Fm(j)p 2017 5553 223 4 v 2017 5637 a Fo(\()p Fn(K)7 b(j)e Fo(\)!)2250 5574 y Fn(X)2325 5588 y Fm(j)2361 5574 y Fo(\()p Fn(z)t Fo(\))p eop %%Page: 7 7 7 6 bop 3809 100 a Fo(7)75 399 y(also)35 b(con)m(v)m(erges)i(in)d(a)h (neigh)m(b)s(orho)s(o)s(d)e(of)i(the)g(origin)e(in)h Fl(C)2143 366 y Fe(2)2223 399 y Fo(and)h(can)g(b)s(e)f(analytically)g (con)m(tin)m(ued)h(in)e(the)75 511 y(sector)e Fn(S)394 526 y Fm(d;\016)518 511 y Fo(with)e(the)i(same)g(exp)s(onen)m(tial)e (gro)m(wth)i(estimates)g(as)f Fn(W)13 b Fo(.)216 624 y(By)31 b(con)m(truction,)g(w)m(e)f(ha)m(v)m(e)i(that)1266 829 y Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))27 b(=)e Fi(P)1722 790 y Fg(\000)p Fe(1)1722 857 y Fm(K)1817 829 y Fo(\()1878 806 y(^)1852 829 y Fn(X)7 b Fo(\()p Fn(t;)15 b(z)t Fo(\)\))27 b(=)2308 806 y(~)2281 829 y Fn(W)13 b Fo(\()p Fn(t)2448 791 y Fm(K)2516 829 y Fn(;)i(z)t Fo(\))p Fn(:)75 1033 y Fo(W)-8 b(e)30 b(conclude)f(that)h(the)f(series) g Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\),)31 b(solution)d(of)h(\()p Fn(E)2062 1047 y Fm(V)2124 1033 y Fo(\),)h(con)m(v)m(erges)h(in)d(a)h (neigh)m(b)s(orho)s(o)s(d)e(of)j(the)f(origin)75 1146 y(in)24 b Fl(C)236 1113 y Fe(2)306 1146 y Fo(and)h(can)g(b)s(e)g (analytically)f(con)m(tin)m(ued)h(in)f(a)i(sector)g Fn(S)2138 1164 y Fm(d=K)q(;\016)2323 1145 y Fk(0)2349 1146 y Fo(.)39 b(Moreo)m(v)m(er,)29 b(there)c(exist)g Fn(\016)3314 1113 y Fg(0)3311 1170 y Fe(1)3377 1146 y Fn(<)g(\016)3516 1113 y Fg(0)3540 1146 y Fo(,)h Fn(C)32 b(>)25 b Fo(0,)75 1259 y(and)30 b Fn(M)35 b(>)25 b Fo(0)31 b(suc)m(h)f(that)1464 1401 y(sup)1388 1489 y Fm(z)s Fg(2)1487 1472 y Fe(\026)1471 1489 y Fm(D)r Fe(\(0)p Fm(;\032)p Fe(\))1692 1401 y Fd(j)p Fn(V)20 b Fo(\()p Fn(t;)15 b(z)t Fo(\))p Fd(j)27 b(\024)e Fn(C)7 b(e)2241 1363 y Fm(M)g Fg(j)p Fm(t)p Fg(j)2429 1305 y Fc(K)p 2390 1313 133 3 v 2390 1349 a(K)s Fk(\000)p Fb(1)75 1640 y Fo(for)30 b(all)f Fn(t)c Fd(2)g Fn(S)540 1659 y Fm(d=K)q(;\016)725 1636 y Fk(0)723 1680 y Fb(1)762 1640 y Fo(.)2969 b Fa(2)75 1976 y Fp(References)75 2179 y Fo([1])97 b(W.)39 b(Balser,)i Fh(F)-7 b(ormal)42 b(p)-5 b(ower)42 b(series)e(and)h(line)-5 b(ar)41 b(systems)g(of)g(mer)-5 b(omorphic)43 b(or)-5 b(dinary)42 b(di\013er)-5 b(ential)267 2292 y(e)g(quations)p Fo(,)32 b(Springer-V)-8 b(erlag,)29 b(New-Y)-8 b(ork,)32 b(2000.)75 2479 y([2])97 b(J.M.)26 b(Bon)m(y)-8 b(,)28 b(P)-8 b(.)26 b(Sc)m(hapira,)g Fh(Existenc)-5 b(e)28 b(et)g(pr)-5 b(olongement)31 b(des)e(solutions)g(holomorphes)j (des)26 b(\023)-44 b(equations)267 2592 y(aux)33 b(d)n(\023)-44 b(eriv)n(\023)g(ees)33 b(p)-5 b(artiel)5 b(les)p Fo(.)32 b(\(F)-8 b(renc)m(h\))31 b(In)m(v)m(en)m(t.)h(Math.)f(17)g(\(1972\),)i (95{105.)75 2780 y([3])97 b(O.)45 b(Costin,)i(S.)e(T)-8 b(an)m(v)m(eer,)50 b Fh(On)c(the)g(existenc)-5 b(e)46 b(and)h(uniqueness)e(of)i(solutions)g(of)f(nonline)-5 b(ar)48 b(evo-)267 2893 y(lution)41 b(systems)g(of)g(PDEs)e(in)h Fl(R)1440 2860 y Fe(+)1531 2893 y Fd(\002)25 b Fl(C)1687 2860 y Fm(d)1733 2893 y Fh(,)42 b(their)f(asymptotic)h(and)g(Bor)-5 b(el)41 b(summability)h(pr)-5 b(op)g(erties)p Fo(.)267 3006 y(Man)m(uscript,)30 b(2003.)75 3193 y([4])97 b(Y.A.)23 b(Dubinskii,)e Fh(Wel)5 b(l-p)-5 b(ose)g(dness)27 b(of)f(the)g(Cauchy)g (pr)-5 b(oblem)27 b(in)e(classes)h(of)g(entir)-5 b(e)26 b(functions)g(of)f(\014nite)267 3306 y(or)-5 b(der)p Fo(.)32 b(So)m(viet)f(Math.)g(Dokl.)g(38)g(\(1989\),)i(no.)e(1,)g (206{209.)75 3494 y([5])97 b(R.)29 b(G)m(\023)-43 b(erard,)31 b(H.)f(T)-8 b(ahara,)30 b Fh(F)-7 b(ormal)34 b(p)-5 b(ower)33 b(series)f(solutions)h(of)f(nonline)-5 b(ar)33 b(\014rst)f(or)-5 b(der)34 b(p)-5 b(artial)34 b(di\013er-)267 3607 y(ential)f(e)-5 b(quations)p Fo(,)32 b(F)-8 b(unk)m(ciala)5 b(j)30 b(Ekv)-5 b(acio)5 b(j,)31 b(41)g(\(1998\),)i(133{166.)75 3794 y([6])97 b(D.)43 b(Gourdin,)g(M.)f(Mec)m(hab,)47 b Fh(Solutions)d(glob) -5 b(ales)45 b(d'un)f(pr)-5 b(obl)n(\022)-44 b(eme)44 b(de)g(Cauchy)g(lin)n(\023)-44 b(eair)-5 b(e)p Fo(.)43 b(\(F)-8 b(renc)m(h\))267 3907 y([Global)30 b(solutions)f(of)h(a)h (linear)e(Cauc)m(h)m(y)h(problem])f(J.)i(F)-8 b(unct.)31 b(Anal.)f(202)h(\(2003\),)i(no.)e(1,)g(123{146.)75 4095 y([7])97 b(Y.)36 b(Hamada,)h(A.)f(T)-8 b(ak)m(euc)m(hi,)37 b Fh(Sur)g(un)g(domaine)i(d'existenc)-5 b(e)37 b(et)h(un)f(pr)-5 b(olongement)39 b(analytique)f(des)267 4208 y(solutions)28 b(de)f(pr)-5 b(obl)n(\022)-44 b(emes)28 b(de)e(Goursat)p Fo(.)g(\(F)-8 b(renc)m(h\))25 b([Domain)f(of)g(existence)g(and)f (analytic)h(con)m(tin)m(uation)267 4321 y(of)30 b(the)h(solutions)e(of) h(Goursat)h(problems])e(J.)h(Math.)i(Pures)d(Appl.)g(\(9\))j(75)f (\(1996\),)i(no.)d(5,)h(469{483.)75 4508 y([8])97 b(Y.)34 b(Hamada,)i(J.)d(Lera)m(y)-8 b(,)36 b(A.)e(T)-8 b(ak)m(euc)m(hi,)36 b Fh(Pr)-5 b(olongements)37 b(analytiques)g(de)f(la)g(solution)h(du)f (pr)-5 b(obl)n(\022)-44 b(eme)267 4621 y(de)37 b(Cauchy)h(lin)n(\023) -44 b(eair)-5 b(e)p Fo(.)36 b(\(F)-8 b(renc)m(h\))37 b([Analytic)d(con)m(tin)m(uations)h(of)h(the)f(solution)f(of)h(the)g (linear)f(Cauc)m(h)m(y)267 4734 y(problem])29 b(J.)h(Math.)h(Pures)f (Appl.)f(\(9\))i(64)h(\(1985\),)h(no.)d(3,)h(257{319.)75 4922 y([9])97 b(S.)34 b(Malek,)i Fh(On)g(the)h(summability)g(of)g (formal)h(series)e(solutions)i(of)e(semi-line)-5 b(ar)38 b(p)-5 b(artial)38 b(di\013er)-5 b(ential)267 5035 y(e)g(quations)p Fo(,)32 b(preprin)m(t)c(a)m(v)-5 b(ailable)30 b(at)h(h)m (ttp://www.mathpreprin)m(ts.com.)f(\(2004\))j(Submitted.)75 5222 y([10])52 b(M.)32 b(Miy)m(ak)m(e,)h Fh(Glob)-5 b(al)36 b(and)e(lo)-5 b(c)g(al)36 b(Goursat)f(pr)-5 b(oblems)36 b(in)e(a)g(class)g(of)g(holomorphic)j(or)d(p)-5 b(artial)5 b(ly)36 b(holo-)267 5335 y(morphic)e(functions)p Fo(.)d(J.)f (Di\013eren)m(tial)g(Equations)g(39)h(\(1981\),)i(no.)e(3,)g(445{463.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0406230407465--