]> www.ginac.de Git - cln.git/blob - doc/polynomial/hermite.ps
* cln-config.1: added manpage, as required by a couple of distros.
[cln.git] / doc / polynomial / hermite.ps
1 %!PS-Adobe-2.0
2 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
3 %%Title: hermite.dvi
4 %%Pages: 2
5 %%PageOrder: Ascend
6 %%BoundingBox: 0 0 596 842
7 %%DocumentPaperSizes: a4
8 %%EndComments
9 %DVIPSCommandLine: dvips -Pljfour -o hermite.ps hermite.dvi
10 %DVIPSParameters: dpi=600, compressed, comments removed
11 %DVIPSSource:  TeX output 1997.02.16:1407
12 %%BeginProcSet: texc.pro
13 /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
14 /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
15 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
16 ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
17 isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
18 hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
19 TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
20 forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
21 /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
22 /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
23 /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
24 string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
25 end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
26 /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
27 N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
28 length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
29 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
30 get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
31 dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
32 /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
33 /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
34 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
35 setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
36 .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N
37 /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
38 gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
39 add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
40 /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
41 dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1
42 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
43 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
44 putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
45 adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}
46 {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
47 adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2
48 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
49 }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
50 length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
51 cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
52 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
53 add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
54 /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
55 known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
56 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
57 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
58 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
59 {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
60 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
61 ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
62 RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
63 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
64 round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
65 rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
66 {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
67 B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
68 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
69 p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
70 a}B /bos{/SS save N}B /eos{SS restore}B end
71 %%EndProcSet
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510 7E153EA27EA36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE00
511 38C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207001FB512F0
512 B6FCA2C648C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070903807E0
513 E0903801FFC09038007F001E3E7EBC26>I<D801FC147F00FFEC3FFFA300071401000380
514 000181B3A85EA35DA212006D5B017E9038077F80017F010E13C06D011C13FE90380FC078
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516 1F8000011600160E1200160C017F5CA280013F5CA26E1370011F146080010F5CA2ECF001
517 01075CA26D6C48C7FCA26E5A01011306A26D6C5AA214FF6E5AA215B8EC3FB015F06E5AA3
518 6E5AA26E5AA36EC8FC2E2C7EAA33>I<B500E0B539E03FFF80A30007903C000FFE000FFC
519 00D803FCD903F8EB03F8F001E0120103015D6D80000060A26D6E13036DD9037E91C7FCA2
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523 07FFFCA30003D9C00113C0C6496C1300017F14FC013F5C6E13E06D7E010F495A6D6C485A
524 02F890C7FC903803FC060101130E6E5A903800FF186E5AEC3FF05D141F140F6E7E81140F
525 EC0DFCEC19FEEC38FF4A7E9138603F8002C07F0101131F49486C7E02007F01066D7E010E
526 1303496D7E013C80017C80D801FC1580D80FFE4913C0B5D8800F13FFA3302B7FAA33>I<
527 B539F001FFFCA3000790C7EA7FE06C48EC1F8000011600160E0000150C6D141C6D1418A2
528 6E1338013F1430A26D6C5BA26E13E0010F5CA26D6C485AA2ECF803010391C7FCA2903801
529 FC06A2ECFE0E0100130CA2EC7F18A215B8EC3FB0A2EC1FE0A36E5AA26E5AA36EC8FCA214
530 06A35CA25CA2123C007E5BB4FC5CA25CEAFE01387C0380D87007C9FCEA3C1EEA0FFCEA03
531 F02E3F7EAA33>I E end
532 %%EndProlog
533 %%BeginSetup
534 %%Feature: *Resolution 600dpi
535 TeXDict begin
536 %%PaperSize: a4
537 %%BeginPaperSize: a4
538 a4
539 %%EndPaperSize
540
541 %%EndSetup
542 %%Page: 1 1
543 1 0 bop 404 573 a Fm(The)33 b(Hermite)f(p)s(olynomials)d
544 Fl(H)1604 588 y Fk(n)1651 573 y Fm(\()p Fl(x)p Fm(\))k(are)g(de\014ned)
545 h(through)1159 801 y Fl(H)1240 816 y Fk(n)1287 801 y
546 Fm(\()p Fl(x)p Fm(\))28 b(=)g(\()p Fj(\000)p Fm(1\))1752
547 760 y Fk(n)1799 801 y Fl(e)1844 760 y Fk(x)1884 737 y
548 Fi(2)1945 801 y Fj(\001)1994 655 y Fh( )2098 734 y Fl(d)p
549 2070 778 107 4 v 2070 870 a(dx)2186 655 y Fh(!)2252 678
550 y Fk(n)2315 705 y Fh(\020)2365 801 y Fl(e)2410 760 y
551 Fg(\000)p Fk(x)2505 737 y Fi(2)2544 705 y Fh(\021)257
552 1022 y Ff(Theorem:)404 1186 y Fl(H)485 1201 y Fk(n)531
553 1186 y Fm(\()p Fl(x)p Fm(\))33 b(satis\014es)h(the)f(recurrence)h
554 (relation)1661 1375 y Fl(H)1742 1390 y Fe(0)1781 1375
555 y Fm(\()p Fl(x)p Fm(\))28 b(=)f(1)1093 1607 y Fl(H)1174
556 1622 y Fk(n)p Fe(+1)1311 1607 y Fm(\()p Fl(x)p Fm(\))h(=)g(2)p
557 Fl(x)22 b Fj(\001)g Fl(H)1831 1622 y Fk(n)1878 1607 y
558 Fm(\()p Fl(x)p Fm(\))g Fj(\000)h Fm(2)p Fl(n)f Fj(\001)g
559 Fl(H)2391 1622 y Fk(n)p Fg(\000)p Fe(1)2528 1607 y Fm(\()p
560 Fl(x)p Fm(\))257 1771 y(for)33 b Fl(n)c Fj(\025)h Fm(0)j(and)g(the)h
561 (di\013eren)m(tial)d(equation)i Fl(H)2018 1711 y Fd(00)2010
562 1795 y Fk(n)2062 1771 y Fm(\()p Fl(x)p Fm(\))23 b Fj(\000)g
563 Fm(2)p Fl(x)g Fj(\001)g Fl(H)2583 1711 y Fd(0)2575 1795
564 y Fk(n)2621 1771 y Fm(\()p Fl(x)p Fm(\))g(+)g(2)p Fl(n)f
565 Fj(\001)h Fl(H)3135 1786 y Fk(n)3182 1771 y Fm(\()p Fl(x)p
566 Fm(\))29 b(=)f(0)257 1891 y(for)k(all)f Fl(n)d Fj(\025)g
567 Fm(0.)257 2055 y Ff(Pro)s(of:)404 2219 y Fm(Let)34 b
568 Fl(F)44 b Fm(:=)821 2153 y Fh(P)908 2179 y Fg(1)908 2244
569 y Fk(n)p Fe(=0)1072 2173 y Fk(H)1130 2181 y Fc(n)1172
570 2173 y Fe(\()p Fk(x)p Fe(\))p 1072 2196 196 4 v 1138
571 2254 a Fk(n)p Fe(!)1277 2219 y Fl(z)1326 2183 y Fk(n)1408
572 2219 y Fm(b)s(e)34 b(the)h(exp)s(onen)m(tial)e(generating)h(function)g
573 (of)f(the)i(se-)257 2340 y(quence)27 b(of)c(p)s(olynomials.)38
574 b(Then,)27 b(b)s(ecause)f(the)f(T)-8 b(a)m(ylor)23 b(series)i(dev)m
575 (elopmen)m(t)g(theorem)257 2460 y(holds)33 b(in)e(formal)g(p)s(o)m(w)m
576 (er)i(series)g(rings)f(\(see)i([1],)f(section)f(2.16\),)g(w)m(e)i(can)f
577 (simplify)1023 2688 y Fl(F)97 b Fm(=)83 b Fl(e)1387 2647
578 y Fk(x)1427 2624 y Fi(2)1487 2688 y Fj(\001)1568 2580
579 y Fg(1)1544 2605 y Fh(X)1537 2787 y Fk(n)p Fe(=0)1715
580 2621 y Fm(1)p 1697 2665 86 4 v 1697 2757 a Fl(n)p Fm(!)1809
581 2542 y Fh( )1912 2621 y Fl(d)p 1884 2665 107 4 v 1884
582 2757 a(dx)2000 2542 y Fh(!)2066 2565 y Fk(n)2130 2592
583 y Fh(\020)2179 2688 y Fl(e)2224 2647 y Fg(\000)p Fk(x)2319
584 2624 y Fi(2)2358 2592 y Fh(\021)2430 2688 y Fj(\001)21
585 b Fm(\()p Fj(\000)p Fl(z)t Fm(\))2681 2647 y Fk(n)1183
586 2925 y Fm(=)83 b Fl(e)1387 2884 y Fk(x)1427 2860 y Fi(2)1487
587 2925 y Fj(\001)22 b Fl(e)1582 2884 y Fg(\000)p Fe(\()p
588 Fk(x)p Fg(\000)p Fk(z)s Fe(\))1822 2860 y Fi(2)1183 3070
589 y Fm(=)83 b Fl(e)1387 3029 y Fe(2)p Fk(xz)s Fg(\000)p
590 Fk(z)1589 3005 y Fi(2)257 3234 y Fm(It)33 b(follo)m(ws)f(that)923
591 3195 y Fk(d)p 905 3211 72 4 v 905 3268 a(dz)987 3234
592 y Fl(F)42 b Fm(=)27 b(\(2)p Fl(x)c Fj(\000)g Fm(2)p Fl(z)t
593 Fm(\))f Fj(\001)g Fl(F)14 b Fm(.)44 b(This)33 b(is)f(equiv)-5
594 b(alen)m(t)33 b(to)f(the)h(claimed)e(recur-)257 3354
595 y(rence.)404 3475 y(Starting)i(from)g(this)g(equation,)i(w)m(e)g
596 (compute)f(a)g(linear)f(relation)f(for)i(the)g(partial)257
597 3595 y(deriv)-5 b(ativ)m(es)33 b(of)f Fl(F)14 b Fm(.)43
598 b(W)-8 b(rite)32 b Fl(@)1323 3610 y Fk(x)1395 3595 y
599 Fm(=)1529 3556 y Fk(d)p 1509 3572 77 4 v 1509 3629 a(dx)1627
600 3595 y Fm(and)h(\001)1898 3610 y Fk(z)1966 3595 y Fm(=)27
601 b Fl(z)2147 3556 y Fk(d)p 2129 3572 72 4 v 2129 3629
602 a(dz)2211 3595 y Fm(.)43 b(One)33 b(computes)1674 3759
603 y Fl(F)41 b Fm(=)28 b(1)21 b Fj(\001)h Fl(F)1601 3966
604 y(@)1652 3981 y Fk(x)1696 3966 y Fl(F)42 b Fm(=)27 b(2)p
605 Fl(z)g Fj(\001)22 b Fl(F)1581 4188 y(@)1637 4147 y Fe(2)1632
606 4213 y Fk(x)1677 4188 y Fl(F)42 b Fm(=)27 b(4)p Fl(z)1983
607 4147 y Fe(2)2045 4188 y Fj(\001)22 b Fl(F)1393 4420 y
608 Fm(\001)1474 4435 y Fk(z)1514 4420 y Fl(F)41 b Fm(=)28
609 b(\(2)p Fl(xz)f Fj(\000)22 b Fm(2)p Fl(z)2133 4379 y
610 Fe(2)2173 4420 y Fm(\))g Fj(\001)g Fl(F)1216 4652 y(@)1267
611 4667 y Fk(x)1311 4652 y Fm(\001)1392 4667 y Fk(z)1432
612 4652 y Fl(F)42 b Fm(=)27 b(\(2)p Fl(z)g Fm(+)22 b(4)p
613 Fl(xz)2050 4611 y Fe(2)2112 4652 y Fj(\000)h Fm(4)p Fl(z)2310
614 4611 y Fe(3)2350 4652 y Fm(\))f Fj(\001)g Fl(F)781 4884
615 y Fm(\001)862 4843 y Fe(2)862 4909 y Fk(z)902 4884 y
616 Fl(F)41 b Fm(=)1110 4788 y Fh(\020)1159 4884 y Fm(2)p
617 Fl(x)23 b Fj(\001)e Fl(z)27 b Fm(+)22 b(\(4)p Fl(x)1647
618 4843 y Fe(2)1709 4884 y Fj(\000)h Fm(4\))f Fj(\001)f
619 Fl(z)2016 4843 y Fe(2)2079 4884 y Fj(\000)h Fm(8)p Fl(x)h
620 Fj(\001)e Fl(z)2403 4843 y Fe(3)2466 4884 y Fm(+)h(4)g
621 Fj(\001)g Fl(z)2734 4843 y Fe(4)2774 4788 y Fh(\021)2845
622 4884 y Fj(\001)g Fl(F)257 5061 y Fm(Solv)m(e)33 b(a)f(homogeneous)h(5)
623 22 b Fj(\002)h Fm(6)32 b(system)h(of)g(linear)d(equations)j(o)m(v)m(er)
624 h Fl(Q)p Fm(\()p Fl(x)p Fm(\))f(to)f(get)1170 5224 y(\()p
625 Fj(\000)p Fm(2)p Fl(x)p Fm(\))22 b Fj(\001)g Fl(@)1550
626 5239 y Fk(x)1594 5224 y Fl(F)36 b Fm(+)22 b Fl(@)1847
627 5183 y Fe(2)1842 5249 y Fk(x)1888 5224 y Fl(F)36 b Fm(+)22
628 b(2)f Fj(\001)h Fm(\001)2286 5239 y Fk(z)2326 5224 y
629 Fl(F)42 b Fm(=)27 b(0)257 5388 y(This)f(is)e(equiv)-5
630 b(alen)m(t)25 b(to)g(the)g(claimed)e(equation)i Fl(H)2121
631 5329 y Fd(00)2113 5413 y Fk(n)2165 5388 y Fm(\()p Fl(x)p
632 Fm(\))7 b Fj(\000)g Fm(2)p Fl(x)g Fj(\001)g Fl(H)2622
633 5329 y Fd(0)2614 5413 y Fk(n)2660 5388 y Fm(\()p Fl(x)p
634 Fm(\))g(+)g(2)p Fl(n)g Fj(\001)g Fl(H)3111 5403 y Fk(n)3157
635 5388 y Fm(\()p Fl(x)p Fm(\))28 b(=)f(0.)1852 5637 y(1)p
636 eop
637 %%Page: 2 2
638 2 1 bop 257 573 a Fb(References)62 792 y Fm(1)146 b(Bruno)46
639 b(Haible:)69 b(D-\014nite)45 b(p)s(o)m(w)m(er)i(series)g(in)e(sev)m
640 (eral)i(v)-5 b(ariables.)82 b Fa(Diploma)46 b(thesis,)501
641 912 y(University)35 b(of)g(Karlsruhe,)f(June)h(1989)p
642 Fm(.)43 b(Sections)33 b(2.15)f(and)h(2.22.)1852 5637
643 y(2)p eop
644 %%Trailer
645 end
646 userdict /end-hook known{end-hook}if
647 %%EOF