
Ʒ/ѧŮ
ԭ/ѧ`롷
/Ǻ
廭/SDwing
/ׯӺ
ͼԴ/ĳX
¼/ù
У/ù
У++TXT/JoyJ from IVocaloid http://bbs.ivocaloid.com
ͼ/ĳXϡù
ͼ/ù
ת ע
Ȫͽ http://www.cnfmp.net/bbs
IVocaloid  http://bbs.ivocaloid.com
ΪͶ ɾϢ

߼
Ǻ (Hirosi Yuuki)
һȤǡдʽ롸д项ϲڻüʱ䲻ϵظĶͬһ顣Ҳϲ֣رϲ͹ԼֵķסҲᴵľѣrecorderϲӰɢǳϲáϲ֣ԡáûʲôȤǸޡСӣΪú˷ʱ䣬Ժȥһˣǳϲ༭ڿ֮ߺȾơ
վ http://www.hyuki.com/index.html

廭
Sdwing
SD
Ϊְҵϡҵǡڰ硹յļһĿǰ¸ֻ滭ӰֻϷ趨ԭϷʿš廭ձ鼮˯µ䡷ϵв廭Ȥǽ˸˫βҶָ̡
վ <> http://blog.yam.com/sdwing

ݼ飺
ϲѧó˽ϸкΪʱһµڿѧ죬һλͬѧȤĲŮ׶ȡ˺˳ڷѧͼѧʽ
һȥǵĹѧõٵǉҲȦӡȻһʼֻΪѧĳɼٵǉҲرѧҲ͸ѧᷢչԵĹϵѧ˵سǰ


TXTʽ˵
a<nη>-----------anηرģ2η<ƽ>ʾ3η<>ʾ
a<nεݽ׳>-----------anεݽ׳
<a>-----------a2θ
<nθa>-----------anθ
a<0>a<1>a<2>-----------±a0a1a2
a<b,c>-----------aϱb±c
رģ()<b,c>ָšڡϱ0±1
<k=mn,f(k)>-----------kֵm֮󲻶ϼ1ֱnkֵf(k)۳
<k=mn,f(k)>-----------kֵm֮󲻶ϼ1ֱnkֵf(k)ۼ͡
lim<xa,f(x)>-----------xaʱf(x)ļ
<,n,k>-----------nԪѡkԪصķ
<For All>-----------С
S<eΪףf(x)>-----------logeΪf(x)Ķ
ͼ

TXT׼ö߰ʽͼԭTXT
TXTĶעһпԱȽ϶ֵĶĶPSPĺ+xReaderСֺTTF塣һʾСôֱȿִܻλ󣬸ĶӰ졣⣬顰ʺϡĻˮƽĶ
УҪʽȫƪԼ50ҡԸ¼еƽ˷롣

ôֻJoyJ
1.ʵϣĳJ뷨鲢һС˵һʽȽرĿնˣѧȤϵͣĶȤۿۡѧRIPӦҲôҶԱһŵӡаް뱣־Ա׻
2.TXTѶ˷ָֹʽͼʵͷۣ൱ӣʡԣͷһμ400kϵķС˵ҡˣɧˡ
3.롶ѧŮ޹ϵ
4.RIPοĶ
֮Τߤˣ
By IVocaloid ֮ͼݳ JoyJ


Ƽ
ں嫵֪ʶУѧһȴѧϰѧʡкܶĸҲиĶЩҪѧѧȴֲܵһζҪȥ˼⡢ȥ߼ܶΪЩԭ֮ȴʧȥ῿Լõ𰸵ɾͣһϧ顣
ʺϱѧȤĶҲʺϴѧѧĹߡеĽɫóѧˣö߸ĽⷨĲõ˼Ŀһ֣̽ͬʱҲϲѧȴֵĽɫĶ߾ͿվһĽǶѭ򽥽ѧϰ˻ѧУ̵ѧ﷦ζӴһ˸Ĺۣݲǹ̶ĳУǹ㷺ȤʽӣǿȽܸֻöᵽͬȤľʦύǿƬȥҪĲǴ𰸣Ǵʽϻոµľ鼰ᣬѧϰֵ
ͬ˵ѧԿԽʱ䣬ЩǧǰѧǸܵȤɾͣλִɿԸͬܣ÷ԼʣܵõµϾõĽűԲṼλøλѧС
һŮѧʦ
־


ҳ
뱾йص鱨ַá
http://www.hyuki.com/girl/
ҳ߸վһ֡



м򵥵СѧĲ֣Ҳѧѵĸ⡣
Ｐͼǳ˼;⣬Ҳʹѧʽ֡
޷ѧʽĺʱȰѧʽһԣרĵܾɣٵٻһͬǰġ
ѧŵĶǣܾ֮⣬ҲעеѧʽôĻᵽڹ±Ȥ


CONTENTS
Ƽ


1¡
11ӣ
12Լ
13ûȷ
2¡Ϊʽ
21Уſ
22
23
24ѧ
25ͽ
251Ķ
252ֵĶ
26ؼҵ·
27Լ
28׶ȵĽ
29ͼ
291ʽʽ
292ʽ͵ʽ
210ѧʽı˭
3¡صĻ
31ͼ
32ת
33صĻ
4¡쳲ɺ
41ͼ
411Ѱҹ
412ȱеĺ
413
414ɺ
42ץס쳲
421쳲
422쳲еɺ
423չʽ
424ʾ
425
43ع
5¡ƽ뼸ƽĹϵ
51ڡѧ֡
52
53ʽ
54һ
55νѧ
6¡׶ȵ
61΢
62
63΢
631һκx
632κx<ƽ>
633κx<>
634ָe<xη>
64ó
7¡ۻ
71ͼ
711׶
712ٵ
713ƹʽ
72ڻؼҵ·Ͻ廯
73ڡBeansʽ
74ԼнɺĻ
75ͼ
751׶ȵĽ
752ɺ
753Χ
754Ĺؿ
755
756뾶ΪԲ
8¡
81Ѱ
811ٵ
812׶
82Իеͼ
821ֺ
822Ȼĵطʼ
823
824ȫ
825Ǵڵ
83ֹݵֽ
84Ħ
85޴Ĺ
86ڽ
87磬
88֪Կףδ֪
89ֻĻ
891ۻ
892ȱȼ
893ӷֽΨһ
894޵֤
810̨
9¡̩չʽ뱴
91ͼ
911ſƬ
912ʽ
92ѧϰ
93Beans
931΢ֵĹ
932΢
933sin x̩չʽ
934ļ
94Լ
95
96ͼ
961ٵٵĳ
962ȥδ
963޵ս
10¡ֲ
101ͼ
1011ֲ
1012˼ʵ
102ȥ·
1021쳲
1022
103Beans
104Լ
1041Ϊѡ
105ֽ
1051ҵķ(ֲɺ)
1052׶ȵķ(ֲϽ)
1053ٵٵķ
106
107ѰҸϽ;
1071ɺ
1072һתǡɺ
1073ߵɭ̩֡չʽ
1074ߵɽ𡻵
1075;յ
1076ٵٵĻع
108ټ
β

οĶָ






ֻǼ䡣
޷䡣
С

޷ǡ
޷ڸʱһоѧŮǡ
ŵĽⷨĵĲŮ׶ȡ
ʵĻŮٵ١
ÿǶʱ⣬оͻḡֳѧʽչ˼άѧʽԽʱ䣬չŷá˹Լѧҵһ
ѧǳԽʱġ
ĶѧʽƷŴǰѧĸжʹǰͱ֤ҲνڵȷʵӵЩ߼ĿС
ѧԽʱ䡣
ҳصıءѧǸ˷ܵϷѵĽⷨΪĿ꣬ľѧļµս
ʱҿʼʹΪѧȴ޴Կƣ޷Լ񣬾޷ƶǵ˼
ֻǼ䡣
޷䡣
һеĿˣڸһ괺졭



1¡


һһ
һǶ
uӡd·ǡ


11ӣ

һ괺졣
ѧա
ӣʢÿ˶ܳոµһ⴫ͳУŬضѧѳɡ
Уݽյҽ磬۾סǷ
ѧYᣬصҵ;Уĵ뿪Уᣬһ̤M˲ŵӣΧ]κ˵·ϡ
15ꡣ151617ҵʱ18ˣᾭһ4ıһ

1535
162222244ı
1717
18233232


ڽͬѧӦҽܰɡóҽˣҪ˵ʲôԼأ
ϲѧȤʽλָ̡
˵ʻôĿɿڴɡ
ˣ͹ʱһΣȻͼȹʽɡ
ʱһø޴ӣҵǰ
һλŮվӣ
Ҳһ͵ܳ
Ҳ̧ͷӣ谵ɫӳ
һ紵ӣŮΧס
Ůҡ
һԼںڵĳ
촽ϴŽ۾
ķ
һһ

1123

ĸ֮Ů첢ָңƺڶ˵㣬㡣شһʲô
ҵҲָԼ
(Ҫһش)
ŮԵصͷʳָȻָҡ
ô£ΪʲôӣұҪֲֵϷأǡ
1123
ţԭˣҶˡ
1123֮5813Ȼ21ǡ
Ůĳңֹͣơ

һ⣬һĸ֡

1427256

Ůָҡ
ǲ
1427256
˲ҵ˹
14272563125ɡû취㡣
ŮüԻشû취㡻ҡҡͷȻ󽫴𰸸ҡ
1427256312546656ʮ
ŮۣҪӣ̧ͷʳָϵڿбȻ
ŮȻͣ֡Ȼֻӽ΢СĶҵĿȴѾ벻Ůʲô
Ȼ

6153577

ĸ֡
6153577
⻹ѵģŬͷת6153ı35ȴǣ35777ıҪдֽϵĻѸٽ⿪ˡ
΢һ¡ӣµŮȻվԣ൱ı鿴ңͷմӣҲΪ⡣̬ȣȻǲ
֪ˡ
ҲŸ˵꣬Ů۾Ϊ֮һ¶Щ΢Цݡҵһο΢Ц
6153577֮133ҲԾ
Ůȴҡҡͷ¶һû취ı顣ʹĳ裬ҲӣƮ䡣
󡣡Ůָ۾
󡭡ȷˣ1113143Ŷԣ133
Ůʣ

62821018

΢һ£18ͷʹҪ2Ļͺˣ⿴ϡԡȫż𣿡Ҷˣ
ٹ412106Ǹֵ⡣˵
𣿲㲻Ҳˣ
¶ͬʱߵҵǰһָֻ֡൱ϸ
(Ҫ)
ڸ㲻״º֣˫˳ů֡
׶ȣָָ
׶ȵˡ


12Լ

ҹ
ϲҹ˳˯ҵʱ䡣ûκ˻ŵ磬ֻһˣ̯鱾̽硢ѧĴϡеĶ峺ĺˮ׳ľľ޷ˡ

׶ȡ

ֻǵһμ棬ȴ̸ֻĹˡһϲѧɣȫûκ˵£ͻȻͬѧʴҺϸһ֣Ƿǳ֣Щ΢ζǵŮ㡣

Ů

ҽ۾ϣűۻ׶ȵĶԻ
һʼ11235811쳲С11֮󣬽ǰγһ

111121232353585813

һ1427256312546656ⷨǡ
1<1η>2<ƽ>3<>4<4η>5<5η>6<6η>

Ҳ˵һΪn<nη>4<4η>5<5η>ã6<6η>ĻҾ޷ˡ

6153577143ֹ򡭡

2335577111113

Ҳǡһ1113ļҵİܱʣ׶ȷ˵󡻻ǲ
⣬62821018412106൱ѣΪʮԲʦеĸλ2֮ɵС

У3.141592653Բ
3141592653λ
62821018412106λ2

Բ3.141592653޷𣬲޷
䡣
ϲѧΪ䶫Ҹϲ˼ѧҪ¾ɵļ䣬Ҫչµķ֡ԵĶֻǣ֡Ԫرûеڶַѧͬ񽫲Ϻ͵׼÷ϣʤĹؼǼ䣬˼
ôġ
ǣûô
ͬʱҲע⵽Ϊʲô׶ڳ6282ʱûֻ˵6282˵62821018ԭˡֻ˵6282ĻͲһֻЦиλ2ֿԣ򵥵Ľ628210ȻҲпһҲÿżвNС

6282102122

׶ڿǵ֮ġ
㲻Ҳˣ
Ԥа취𣬻¶ı顣
׶ȡ
ڴӣķУѷ羰վҡҷĺڷָӼҰϸָԼ˫ů뵭ζ
֪ΪΣҵԺѾӲȥӰ


13ûȷ

׶ȣΪʲôʱҪأʵ
ʱ򣿡ֹͣ㲢̧ͷ
ͼң򿪵Ĵˬķ磬һƬ̣ͩԶϰ
¡
ѧУ½ҡͬѧʸ𽥵һصƽճϢ
ûвμκţҲνĻؼ硣ˣѧҲ̻ؼҡİΪܶʽͨǰͼҡ
ͺ͹ʱһûвμţǷѧͼ(гΪͼ)飬ǿɫֻԤϰ븴ϰΡ
ϲĻʽϳֹĹʽڱʼǱϡԭĶ廹ԭµʽΡʵƷ任Ȥ˼֤ʽϲЩڱʼǱͷ
ó˶ҲûпһѵңȤһ˶ԱʼǱʱȻдʽңǲôдԣһĹйһϷûбܡɵϷˣʷϵѧսϷֻһ֧ԶǦʡһʼǱҵԴܽеϷִ˲ƣ
ԼʹΪҲͬһͼҵȤ
ǣʵԭЩͬ
Ǿǵͼҵѧֻһ
׶ȡ
ͬ࣬ÿڷѧͼһΡ
ÿһ˼ʱὫԶǦʴҵгȻڱʼǱд֣˵һ£ⱾʼǱҵģ˵˻أ
ҲֳѧȻѣȤ̼ҡ
˵ʱָʲôʱ򣿡׶ҧ(ҵ)ԶǦʻš
ǵһμʱӣ¡
ǸûʲôɣֻǸպ뵽ˡôͻȻ
ûʲôͻȻ뵽ѡ
ϲ𣿡
Ӧòᡣ
ร֪ûȷ仰𣿡
ʲô˼
Ʃ1234Ϊʲô׶˵
Ȼ5ɡ12345ȥ
ȻⲢһġ1234ȻͻȻ10203040ӵ100200300400ҲһС
̫ˡʼֻ˵ĸȻ֮ų֡ͻȻӡ֮ġ˭Ԥ⵽1234֮10
Ҫ㼸ſأһֱĻҪʾڼŹأ
νġûȷ𡻾˼ʾ֮пͻȻıɡ1234ͻȻ10Ļ˵δ̫ûˡ
ϵ²ôһ𣿲֮ᷢʲô¡ԭԤĲͬôܽеһ𣿡
׶һ˵һ߽дڱʼǱϡ

1234698121827

šֺ֪֪
1234֮Ӧ5ɡǲ56йûг֣Ҳ˵ɡ
š
123469Ӧûɣ෴9һǱȽС8ԭûԽԽȴˣܿĹ𣿡
šһʼ1⣬ֵĶ23ıСĲ־Ͳ̫˽⡣
ĻͿ˵ˡ

2<0η>3<0η>2<1η>3<0η>2<0η>3<1η>2<ƽ>3<0η>2<1η>3<1η>2<0η>3<ƽ>2<>3<0η>2<ƽ>3<1η>2<1η>3<ƽ>2<0η>3<>

23ָ˼ĻͿԿṹˡ
ףһǲ̫0η1ԡ

2<0η>3<0η>12<1η>3<0η>22<0η>3<1η>3

ȷʵһУ
šдָ˻ǲأ
 2<0η>3<0η>-----------ָĺ0
         2<1η>3<0η>2<0η>3<1η>-----------ָĺ1
         2<ƽ>3<0η>2<1η>3<1η>2<0η>3<ƽ>-----------ָĺ2
         2<>3<0η>2<ƽ>3<1η>2<1η>3<ƽ>2<0η>3<>-----------ָĺ3
ԭˡ
˵23ı׶˵һ롣
ʱͼҵſͻȻ
׶ȡ˰ɣ
ϰ𣿡
׶ȰѱʻҲڵŮȥҪ߳ͼ֮ǰתͷ˵
һٸֻۡȤɡ
һ뿪ͼҡ
ټֻ
ô£





2¡Ϊʽ


ҵֻ
һβ`˩`


21Уſ

ϸ߶ֻѧ־IIIԾɺһûб仯֮ǰôΪġ

롢

µף߶󾭹һµϣУſڱһŮס
һɫţҺͿţŮи󣬾Уķߡ
߱Ұܶ࣬Ҳûпӡ󣬴ǸѧҼæŴȥ
ϴյŮСѧʱʱҸðϢΪ೤ŮҵԼҶڵ㣬ҪѧУϿร͵ֻǵ
֮ǰ׶˵֮ᷢʲô¡ȷûδغһģһ
ڴŲڿжҡҵġ


22

⡣1024м

ʱ䣬Ůóʱ׶ȱҧɿߵҵ⡣ûл࣬Һ׶ȶ꼶Ȼͬࡣ
㣿ҰŷŻؿڴ
10֮ǰش0123
һ£1024ӡ10241ԣ2ԣ3У1024ܱ34Ļԡˣ1024210ηҼææؼ㡣
910ʱ䵽
11102411
ȷ𰸣ôģ׶һɿָһߵһش
1024ӷֽĻ210η1024ͻֽ⡣˵

10242<10η>2222222222-----------210

ҽ˵1024һ1024һ2nn1101024Ϊ11
2<0η>2<1η>2<ƽ>2<>2<4η>2<5η>2<6η>2<7η>2<8η>2<9η>2<10η>

ҵĻش׶ȵ˵ͷˣôһ⡣1024ȫĺǡ
׶ȣǸе£ġ˵֮վ
Ҳ˱ϻ⡢¶׶ȣٵ߳ҡ
ĺܱǸ1024ӵĺͰұݶ˼𰸡
	

23

ʹݣݶϻûʲôˣΪ̫õĹϵɡ
ŷװŰɫֽøֱʺдľ塣

ǽѧĵٵǉǸѧͶͬССһѧáΪѧ۹ѧ£дš
ȻҶѧȤеϿݾҺܳ˵֮ѧϣܽ⡣
ǳǸæµʱˣϣл̸ѧһڴͽ
ٵǉ

ҽŶĴΡ
ԭٵǉĦŵ===ٵǉͬССһѧáȫûӡ󣬲óѧѧȷʵܶ࣬
ȲЩŻõȻеʧˣҲá
ѧڴͽҼ氡


24ѧ

𣿡

һĿγ̣ǰͽҵ;У׶ͻȻʡ
2047̻ش1024ĺ;2047
Ϊ˼ʱԣɡ
š
Ҫȥͼ𣿡׶ȵ۾һ¡
ŲȥˣͻȻе¡
มĻͳؼҵɡ

׶ȳĻؼҵ
˵һn䡸͡ķΪΣ

Ҫʹn͵ʽ𣿡ʡ
ֻҪдĹ̾ͺˡ


25ͽ

Բ
սͽңͿٵһ˽ŵصţǰűʼǱǦʺС
ҡѧ̸һ̸ֲ֪ô졣˵ĻȽϷ㣬ԡ
ͽұУƹССͥܵҪͻѧʱʹáɽݹɣ½ǽ̨Ϊѧ㿴ʵá
Һ͵ٵһŵĳϣҴӿڴóϵš
Ѿˡǲ˼Ҳ̫ǵ㡣
ǰҡΡ
ûϵҲ㲻ǵҵġ
ңΪʲôʶҰڹʱӦòô۲ŶԡûμšѧֻͼӦòעĿ
ѧมҡǸ
ˡ˵̸̸йѧֵ飬˵ϸһ𣿡
õģллҴСѧʼ;ѧȤǽк󣬲Ͽλǿαá޷ȫ⡻֮ʦ˵ѧҪҪúѧϰԲҪ޷ȫ⡻⡣
ԭˡԾΪС޷ȫ⡻⣬ĳɼҲǺܺÆ
Ļ
ٵٱ߽ʳָָ׷촽ϱ˼ӵһͷ̷˫۵ĵٵǉ˵ĸоǻõССè֮ġ
ο֮һΧĿԣͲ̫⡣ģ⿼ʱͻῼúܲм
ϿأϿεʱö𣿡
Ͽΰʦ̵ʱˡ
ȴ޷ȫ⣿
ǰ޷ȫ⡣ܽ⣬ϿҲöʵȴûȫ⡣


251Ķ

ôʵøһ㣬֪𣿡
ţӦ֪
Ӧ֪˵˵Ķ壬ǻشʲôҪʽԼ˵Ϳˡ
ʲôš57֮𣿡
ţ57û57ֻһӡ͡塹һʲô
õġǡֻ1ԼԼɣѧʦһҪĶ塣ٵٵͷ˵
Ҳ˵Ϊȷģ

pֻܱ1pʱpΪ()

ţҾȷġ
ⶨǴġ
ף5ӵĻֻ15
ţ5ûĻ1ҲˡΪp1ʱpֻܱ1pǷϵģ1֮ڡС2СУһ2ʼ

235711131719

Ҽ˵ȥǰĶǴģĶӦд

pֻܱ1pʱpΪ1⡣

ǴһʼͶ

pΪ1pֻܱ1pʱpΪ

ʽҲԡ

p1pֻܱ1pʱpΪ

1ȷʦҲ̵ģܶѧдĶˡǡ
ٵͻȻʰͷ
֪ˣ1һǲͬΪʲôܰ1أȥʲôĵطҲܰ1rationale
rationale
ɡԭ˵۵ĸݡ
มŮҲ֪ͬɵҪ԰
ѧ
ǸΪʲôܰ1أܼ򵥣ΪӷֽΨһԡ
ֽΨһԣΨһʲô
νֽΨһԾָһnӷֻֽһ֡˵24ӷֻֽ2223һ֡ﲻֵ˳22323222֮࣬Ȼ˳ͬȻΪͬӷֽ⡣ӷֽΨһѧ൱ҪģΪҪʣԾͶ1Ϊ
ΪҪʣΪԭͿԶ𣿡
ԵġȻ˵еšѧһҳԹѧõѧȻǶ塣ع涨ǿǶˡǣԶܲó¡Ķ1ʹӷֽΨһʧ˵㶮ӷֽΨһ𣿡
ˡɡ
šΪʲô˵ɡȷԼǷСرǿˡԼ
ҪôȷԼǷأ
ٸʵȷǷˡԽʯȻǶ壬ʵؾҲһֺܺõϰ

1ӷֽΨһ޷

ԭ124ӷֽ⣬ͻкܶ֡

 2223
12223
112223
	.
	.
	.
ǵġӷֽΨһ޷ӡ
ҵĻõٵһ
˵ܶ֡á2ϡķʽ֡Ϊ
ΪȽܣٵϽȥ
ûֱܶ֡﷽ʽܡǺܶࣿ߾ͺģ
ѧƺҲҪһҵԴˡڡ塻ӷֽ⡻ΨһԡܵıѧôҲǺҪأ
ûܴѧǺҪġҪܱᣬѧŻʹܵܵԾʽ
ʽ
ôѧԡʽĻɡΪҪõڰ壬ǵȥ
ͽҵǰٵں棬Ÿ߼һҵıܵһ
ۣ
ԡԲ𡣡
ٵٱ¥ݰײҵı˿ҪʱվȽŲΣա






252ֵĶ

ôֵ֪𣿡ڰ岢վš
ţӦ֪5ľֵ55ľֵҲ5ȥžͺ˰ɡ
šôдxľֵ壬Խ𣿡ںڰʽ

xľֵ|x|Ķ
  |x|x (x0)
 |x|x(x0)
ʾĻҾ뵽ˡȻxľֵѸõxǺ𣿡
ѸõѧǺ˵Ȼܹ˼ҲϷ϶塣
ôѸıġأ
һôxľֵʲôںڰдʽӡ

|x|

ΪҪѸȥxɡ˵|x|x
ԣx3Ļأ
ףx3ĻٵҲںڰ㡣

|x||(3)|Ϊx3
|3| ԣ(3)3
3|3|3

 ˵|x|xĻx3ʱͻ|x|3.ʵ|x|3ԲŻ|x|x
ҵ˵źڰϵʽӣٵϸϸ˼
ԭˣxҲпǸģ״ĻͻҿxͲԾ뵽35֮ˡ
ǰΪxǰûκηűʾͨ뵽xڣ3֮ĸȴҪرʹxǱʾʹúܶʵ˵ҲԾضֵֵǰѸõ˵̫ͳһȷϲУܻëܵ˼ǱҪģϰܵ˼ϰʽѧҲ˵
ٵǰ˸λ£һָŪʼǱԵһ߳˼
ڵȴķԡ
ҵĹ˷ˡ
ô˵
ұһùġҲܵضαĶʽҵѧһúɢɡ
̾һֳһʧӡ
˵㰡
ףٵٿҡ
ôĻڿʼͺˡȥѾȥˣǻڰڷֵδıͿˡ
ٵͻȻ۾Ȼվ
ǡǰڹȥҲûáδͺˡлл㣬ѧ
ţȵΪֹɣɫҲˣ´ٽ⡣
ģ
ţҷѧ󲿷ֶͼңʲôʵģҾͿˣٵ١
һ˲両ֹԣܸ˵¶΢Ц
õģ


26ؼҵ·

ѽˡ
߳Уſڵĵٵգܲտʼꡣ
ûɡ𣿡
ȻпԤϳ̫ϣԾˡûϵֻС꣬ܵľͺˡ
վǰͻʪˡ˳·ҵɡҲһ߰ɡ
˼ллѧ
ƺҵһθŮӹһɡ͵Ĵ֮УȻе㲻ϰߣĲس˳еֵһƬš
һʱ䣬о죬Ҳû뵽ȻиԼĿɰѧã͵ٵɣıͿ֪û
Ϊʲôѧ֪أ
֪ʲô
ǡǽ̸ЩѧйҲĲ֣Ϊʲôѧ֪ﲻأ
ŵˣһΪٵٻĵӦ
Ϊ̸⣬Ҳ;ֵĲ֣ҲʡѧʱΪĵطеš˺þá˺֮ܶ󣬲ŷ֡ԭˣ൱飬ۻЩ󣬻ὥѧȤ֡ǰתɡ
ת䡣ǡThe Bend in the Roadɡ·Ҳܵվ𣿡
ţת䡢סլȽϿ쵽վ
ȽϿ쵽𣿡
ǰϴҲȽϿม
ףٵٵٶͻȻ̫𣿹ȻҪϲ̫ס

˳վ
Ϊҵһ»Ҫȥ˵ټˡˣɡȽɡ
͵
ţ
ûûʲô顣ɡҾˣһỹģлл㡣
ٵٽַǰؾϹ


27Լ

ҹ
ڷٵٵĻȴг徢֮ӦûɳɣҪ֪ѧȤͺˡ
͵ٵ˵ʱҰڳǽ̵ߵ̬׶˵ʱкܴĲͬ׶ʼնһֱ˵һ̵ֱɡ
ó׶ȳĻؼҵȻͬͬѧؼҵ

׶ȵĻؼҵ
˵һn䡺Ӻ͡ķΪΣ

ֻҪnȫҳͺˡҳٰ֮ǼͳɡӺ͡ˡֻشδ̫ȤѰҸһĴ𰸲СţȽnӷֽ⡣
ʱ10242<10η>⽫Ŀ΢廯Ƚnĳݱ֡

np<mη>pΪmΪ

n1024ʱʽp2m10ͬķ˼1024¡

1pp<ƽ>p<>p<mη>

np<mη>״£nġӺ͡¡

(nӺ)1pp<ƽ>p<>p<mη>

ϣܻشnp<mη>Ӻˡ
֮ٹ廯ûôѣֻҪúӷֽһд
nͨӷֽ⣬pqrΪabcΪ

np<aη>q<bη>r<cη>һ£

һ£ʹӢĸĻ޷Եıָ֣abcʾĻܿͻᵽpqrˡʹʽûҡ
Ҫ2<>3<1η>7<4η>13<>ʽҲ<η>ĳ˻д
ȻˣǾôp<0>p<1>p<2>ʾָa<0>a<1>a<2>a<m>ʾԱ0123mдȻʽúܸӣǿԵı֣m1nӷֽʱĸԸĳд

nӷֽ⣬p<0>p<1>p<2>p<m>Ϊa<0>a<1>a<2>a<m>Ϊ

np<0><a<0>η>p<1><a<1>η>p<2><a<2>η> p<m><a<m>η>

nΪṹʱnбʾ

p<0><b<0>η>p<1><b<1>η>p<2><b<2>η> p<m><b<m>η>

b<0>b<1>b<2>b<m>Ϊҷ

b<0>0123a<0>һ
b<1>0123a<1>һ
b<2>0123a<2>һ
.
.
    .
b<m>0123a<m>һ

ţȻдú൱˵ӵָ012һֱ仯ˣͨҪ廯Ҫܶ
廯ͼˣֻҪȫһ͡

(n)1p<0>p<0><ƽ>p<0><>p<0><a<0>η>
1p<1>p<1><ƽ>p<1><>p<0><a<1>η> 
1p<2>p<2><ƽ>p<2><>p<2><a<2>η>

1p<m>p<m><ƽ>p<m><>p<m><a<m>η> ()

ԲԣⲢǡȫӵĺ͡аӳеֺ͡ʵʵӦ񡭡

p<0><b<0>η>p<1><b<1>η>p<2><b<2>η>p<m><b<m>η>

ӵĳȫ֮ѡϲһȷĺ͡˵⣬ʽɡ

(nӺ)(1p<0>p<0><ƽ>p<0><>p<0><a<0>η>)
 (1p<1>p<1><ƽ>p<1><>p<0><a<1>η>) 
 (1p<2>p<2><ƽ>p<2><>p<2><a<2>η>
 (
(1p<m>p<m><ƽ>p<m><>p<m><a<m>η>

׶ȵĻؼҵĽ
nӷֽ⡣

np<0><a<0>η>p<1><a<1>η>p<2><a<2>η> p<m><a<m>η>
p<0>p<1>p<2>p<m>Ϊa<0>a<1>a<2>a<m>ΪʱʽnӺ͡

(nӺ)(1p<0>p<0><ƽ>p<0><>p<0><a<0>η>)
 (1p<1>p<1><ƽ>p<1><>p<0><a<1>η>) 
 (1p<2>p<2><ƽ>p<2><>p<2><a<2>η>
 (
(1p<m>p<m><ƽ>p<m><>p<m><a<m>η>

ܲдøһأšǴû


28׶ȵĽ

ȷȻܸӡ
׶ȿҵĴʱܸɴ½ۡ
а취дø𣿡
ԡ׶̻ش𣬡ȣں͵ĲֿʽӴ棬޶1x0״¡׶һ߻شһҵıʼǱд¡

1xx<ƽ>x<>x<nη>(1x<n+1η>)/1x

ԭˡ˵ǵȱе͹ʽ
ϾͿ֤׶ȼ˵

               1x<n+1η>1x<n+1η>ͬһʽ
(1x)(1xx<ƽ>x<>x<nη>)1x<n+1η>ʽӷֽ
            1xx<ƽ>x<>x<nη>(1x<n+1η>)/1x ͬ1x

һдĳݲֵĺ;ȫɷˡȻĲ־áǡ
ǦеĴд˵
ԣԲûϵ(Product)ǡ(Sum)ĳ˷ʽֻǸպû(Product)ĵһӢĸPϣǡǶѣͬ(Sum)ĵһĸSĲǦһǵĶʽ¡׶ȼ⡣

 <k=0m,f(k)>=f(1)f(2)f(3)f(m)ʽ
áǵĻĲ־ͿԼرʾ˵

׶ȵĽ
nӷֽ¡

n<k=0m,p<k><a<k>η>>

pkΪakΪ
ʱʽnӺ͡

(n)<k=0m,(1p<k><a<k+1>η>)/(1p<k>)>

ԭˣȻ̣Ҳˡ˵׶ȥͼ𣿡ʡ
ᣬҪȥӢӢϰˡ


29ͼ

ѧ㿴ҴӹеĿαѶȫˡҾͿԼϰˡ
ٵҵͼѧңЦ̯ʼǱ
มȻһϾˡ
Һϲรֱһ¿αһκҲŷ֣ѧкܴĲͬ㣬ʽǷ֣԰ɣѧ


291ʽʽ

ô˵ѧʽĻ⣬̸̸ʽʽٵнʽɣ

x10

еģx1ɡ
ţx10ʽͽ⿪ˡʽأ

2(x1)2x2

õģʽ㿴

  2(x1)2x2Ŀ
 2x22x2չ
2x2x220ٽұ
 00

ף00ˡ
ʵ2(x1)2x2ʽǷʽǺʽߵ2(x1)չͻɺұߵ2x2һҲ˵xκʽΪԶԽʽȷ˵Ƕxĺʽ
ʽʽһ𣿡
һνķʽǡxĳʱʽʽǡxκʽԳ൱ͬɷʽĻǡôʽĳ⣬ʽǡʽǷκԳҪ֤ǷΪʽ⡣
ԭԭˣ֮ǰûע⵽ֲ𡣡
ţͨǲע⵽ģעһ»ȽϺãϾ󲿷ֵĹʽԺʽʽ֡
а취һʽ֪ǷΪʽ𣿡
ʱʱУʱҲжϣҲ˵ȥждʽ˵ҪдʽǺʽ
дʽˡ

һʽڱ仯ĹУǺʽʽ

(x1)(x1)(x1)x(x1)1
 xx1x(x1)1
 xx1xx111
 x<ƽ>xx1
 x<ƽ>1


һֱõȺӣxκʽȻҲǱһĺʽһһܵõʽ

(x1)(x1)x<ƽ>1

ԭˡ
һĺʽΪҪŰʽı仯һ֣ԲҪСöʽมָ뷨һһ˽ͺá֪ԺԿʽ

x<ƽ>5x6(x2)(x3)
 0

Ⱥ֮УһȺŹ˺ʽҲǡx<ƽ>5x6(x2)(x3)xԳڶȺǹɷʽʽȫ(x2)(x3)0x<ƽ>5x60˼
มԭҪⰡ
˷ʽʽжʽһӵʽӳʱһ֣ʽӣҪֵʱʹõȺţʽ޷񷽳ʽԽ⿪ҲʽһҪֻ֤ҪԼͿˡ
νĶʽԾٸ𣿡
Ʃ罫е㸴ӵʽӦ(Alpha)(Beta)s֡νҲǶ塪ʽӡ

s¡ʽ

ѧ⣡
ٰõؾ֣ھܽʹþҲûϵǸȤŮ
ѧѾ첻ˣΪʲôҪsأ
ʵʲôԡֻȡ֣stУ㶨s֮󣬺ҪʾʱֻҪsͿԣö壬ܽʽֵ׶
֪ˣǦͦʲôأ
ţָڱĵط֡дsµʱһָõȺߵȺұߵʽҲ˵ڶæͦ¹ɵʽУsȡ
ʽʲôֶ𣿡
ǵģʲôԣǲѾ˼ķš˵Ѿs£ֶ֮s£Ķ˾ͻˡ
˵Ҳǣûˡ
УʹóֵķţԲʵĦлλiȵȣҲú֡ʽгµķʱȱ𼱣롺ǲǶʽأг񣬡sΪ¡ǡʹΪs֮˵ǾһǶʽˡ
ԭ
ǰٵ١ξҳѧкֵĵʽɣ񷽳ʽʽʽӡ
õģһԿ
ѧкܶʽЩʽĳΪ˴Լ뷨дµģЩʽıһдЩѶϢĳˡ
ѶϢĳˡ


292γ͵ʽ

ĶʽʱעʽʽǺҪġ
ʽʲô˼
Ʃʽ

(x)(x)0

ʽǳ˷Ҳǻʽһ˵ɻÿʽΪʽ

 (x)(x)0
   ʽ 
νʽӣʽֽйϵ𣿡
еģʽֽǽʽֽΪʽӷֽǷֽĻʽͨὫ˷ġʡԣ3ʽ˼һģͬķʽ
еģʽֽǽʽֽΪʽӷֽǷֽĻʽͨὫ˷ġʡԣ3ʽ˼һģͬķʽ

(x)(x)0ʹáʱ
(x)(x)0ʹáʱ
(x)(x)0ʡԵʱ

õġ
Ȼ(x)(x)0ʽ֮Уٻһ0ΪʽĽۡ
Ҷˣ˽Ϊ0лһΪ0
ϣһΪ0áٻһΪ0ȽϺãΪп߶0
ٻһΪ0Ҳһܵı𣿡
ûôٻһΪ0ͱʾx0x£0仰˵x¾ʽĽ⡣
õġ
Ž(x)(x)չʽǷʽ𣿡

(x)(x)x<ƽ>xx

ԲԣǺʽ
ܺãչ֮ʹӻɺˣǻ2ʽұǺ͵4

ɺ͵ÿһʽӳΪΪ׶
	תչ
(x)(x)(x<ƽ>)(x)(x)()
	תʽֽ

ʽûеңҪôأ

x<ƽ>xx

ǵģѣx򣭦xһxġ
ǡxҪɡม񣭦x򣭦xЩֻһxΪxһֱӳΪһ
õģxһͱˡ
x<ƽ>()x
		 1
ٵ٣֪ʽγΪͬ𣿡
֪ͬ֮ǰûر⡣
Ҽ⣬һʽǺʽأǷʽ

(x)(x)x<ƽ>()x
(x)(x)x<ƽ>()x

չ֮ͬxԳĻǺʽ
ȷ𰸡ôǰ˼ʽɣǻʽ

(x)(x)0ʽķʽ

ʹøղŵĺʽʽǺ͵ʽķʽ

x<ƽ>()x£0͵ʽķʽ

ʽȻ״ͬȴͬķʽֻúʽߵʽı̬ѡ
ǵġ
ǿʽķʽʱҪ뵽ʽĽΪx£ʽķʽĽͬҲx£Ͼͬķʽ

(x)(x)0ʽķʽ(x)
 
	     
    x<ƽ>()x£0    ͵ʽķʽ(ͬx)

򵥵ĶηʽֻҪÿľܽȽʽǲǳú񣿡
x<ƽ>()x£0(x)
 x<ƽ>5x60

ȷ񡣦»5»6
ԣҲ˵x<ƽ>5x60Ľ⣬ֻҪҳӻ5˻6ɡ仰˵x23
ȷ𰸡
ʽ͵ʽʽʽ֮һ͵ʽ0ʱ׽⣬ǻʽ0ʱһĿȻˡ
Сĸоˣⷽʽ͡ʽкܴĹϵɣ

210ѧʽı˭

ΪʲôѧУʦû취ѧһ̵ôϸأ
ΪҺڶԻʵʱң̻شԲŻñȽϺö˲һһǰҪֻʦϿΣĵطҲʦԭشʦרְ
ٵ˵ȻͻȻ뵽ʲô˵
ѧʱĵطô죿
šһֱĻǺţȻ¶һ֮ٻصԭǺŵĵطһΣٲ¶Ƕ飬غüΣǰô붼벻ʽչھ˼ΪдˣҾѯʣӡˡ
벻Ǻܻʱ𣿡
Ǻܻʱ䡢ǳʱ䣬ǵȻġ뿴ʽһʷڶʽʱ򣬾ǺѧҸ񶷣ỨʱһġչһʽǳԽ˼ʱ⣻ʽʱǶǸССѧҡ
ССѧң
ǰΪҪΪѧҶϸĶʽֻǶҪдʱڻԼǷˣһдȷϡ
ٵٵͷ˷ܵ˵
ѧ˵ġʽԡҲоˣʽıĳҪҵѶϢĳ˻ѧУʦǱдαˣҲǼǰѧҡ֪ͻԽԽѧˡ
ٵٷ·𻳱Ƶ˵롣
˵ٵУſڽסңϣ̸̸
߷ˡšȻ·ૣ
ȻҵıѧĻ
˵һæֵס͡
ҵĻ
û¡ʲô鶼ûС
ٵٵȾһƬɫ

3¡صĻ


ѧıɡ
ж


31ͼ

졣
ĩӣͼʽʱ׶Ƚͼңֱ
תվҵļ±
š
׶ȴŽ۾Ƭһ㱡ɫʶ۾侲ͫס
ֻҪ˼ϵĵλVʽtorƶܶˣûҪǰɡ
׶ȿ˵ۣﳣֱӣһе֣ǰVʽtor
ûϵֻϰ
ҪʽĻΦȵתͺȤม׶Ա£ҵĶС˵ĦӢzetaͷݼĿɦҵĶ䡣
תΣȻʽչ˼תξ͵2ȵתԵõڦȵĺʽ
׶ϵıʣڱʼǱҶСдʽӣͬʱ׶ȵҲҵ֡

cos2ȣcos<ƽ>ȣsin<ƽ>
sin2ȣ2sincos

ʲô
űʼǱϵĹʽлش(ǹʽ)ȴûг
֪ǹʽ
׶վŵ˵㡣
ʼʦ̬һشͼ˵ȥһֱˡ
Ƚǵתʾʽӡ׶˵

     򡡡򡡡

Ƚǵתʾʽӡ
|cos    sin|
|                |
|sin    cos  |

Ƚת2Ρ൱ʽƽ
|cos    sin|<ƽ>   |cos<ƽ>ȣsin<ƽ> 2sincos|
|                |       = |					  |
|sin    cos  |         |2sincos  cos<ƽ>ȣsin<ƽ> |

ԡȽת2ΡΪת2ȡʽӾ͵ʽӡ
|cos2    sin2|
|                  |
|sin2    cos2  |

ȽʽݣԵõʽ

cos 2ȣcos2ȣsin2
sin 2ȣ2sin cos 


Ҳ˵cos 2sin 2ȿcos sin ȱ֣2æȱʾʽӣͳΪǹʽתʽֲ½еݣͿԵǹʽ
õȺűʾ2תһΡ롺ת2Ρ̬ͬʵͬĶʱһõ鰡

     򡡡򡡡

׶˵ͬʱҵԴ˼һ顣ŮŮע⵽ʵͬһʱһõ鰡
ȻԾɲһĬĬ׶˵


32ת

Ȳ֮ǰʽ׶ȱ˵ҵıʼǱд⡣

31
nʾһa<n>

n  01  2345  6  7
a<n> 10101010

ó𣿡
ܼ򵥰10һ1֮أ˵񶯱ȽϺãһش
รԭеġ
𣿡
뷨ûô뽫񶯡һֳ
һҲ˵nʾa<n>˰ɡţ״Ļдˡ
a<n>=1	(n=0,4,8,4k,)
    a<n>=0      (n=1,5,9,4k+1,)
a<n>=1     (n=2,6,10,4k+2,)
ţûͲˡ
׶ȱ˫ۣʳָҡΡ
ô˼⣬Ҫôһأſ۾š

32
nʾһb<n>


n   01  2  345  6  7
b<n>1i1i1i1i

iָ<ţ1>𣿡⡣

λ⻹бi𣿡
ˣȲb<n>nΪżʱǣ11nΪʱΪiiҲ񶯵һ𣿡
Ȼǣ㽫е񶯣
֮⻹бⷽʽ𣿡ʡ
׶ڱ۵һ˲ش
øƽ˼ɡƽxʵᣬyƽ棬ĻȫĸƽԵ֡

     
  xyi  (xy)


32b<n>ø˼Ļ110ii01i

10i01i10i01i10i01i10i01i

bnڸƽԵʾͻͼ

(10)(01)(10)(01)(10)(01)(10)(01)

[ͼƽֱϵĵ(10)(01)(10)(01)]

ԭˡͻΡӦ˵εĶƶһ˵һԲϻߡ

[ͼƽֱϵĵ(10)(01)(10)(01)ʱ˳ֱͷӣ(10)(01)(01)(10)(10)(01)(01)(10)ռͷһ]


ร㽫ͼΰȷʵҲԡ
֮⻹бͼ𣿡ʡ
ԴϵӲأͼأ׶Ȼش

[ͼƽֱϵĵ(10)(01)(10)(01)ʱ˳ͷӣ(10)(01)(01)(10)(10)(01)(01)(10)ռͷһԲ]

Բ
ûԲ뾶Ϊ1ĵλԲڸƽԭΪĵĵλԲǽбֳɵλԲϵĵ㡣
λԲ
ͨλԲϵĵĸʾ

cosȣisin

ˣȾǵλ(10)תǰ

[ͼƽֱϵ(10)ڵһһ(cos,sin)(cos,sin)ԭ㣬xɵĽΪǦȡ]

ûǳƦΪǣĶӦϵ񡭡

                 
  cosȣisin  (cosȣsin)

32ΪΡԲĵȷֵĵ㡣Ҫαĸȷֵأ׶ȶ˵
Ϊ90ȡҲ˵ÿȦ/2ӾˣΪȣ0/2У3/2Ҳ˵ĸΪԲĸȷֵ㡣һش
cos0(/2)isin0(/2) 
cos1(/2)isin1(/2)
cos2(/2)isin2(/2) 
cos3(/2)isin3(/2)

ûһbnһͿԱʾʽӡ׶˵

32
bncos n(/2) isin n(/2)(n0123)

Ȼٻص31an
 a<n>10101010
˵an101񶯰ɣʵǸҲһ˼ʽ

31
a<n> cos n(/2)(n0123)

סΪʲô
ͼ˼ŽղbnĸȷֵͶӰʵᣬͿԿ񶯵ӣ˵תӰӡ

[ͼƽֱϵĵ(10)(01)(10)(01)ʱ˳ͷӣ(10)(01)(01)(10)(10)(01)(01)(10)ռͷһԲΡ֮ڴͼ»ʵᣬ(10)(10)(01)ֱ߶ͶӰʵ]

a<n>кֲܶͬĿԵɡСǡʵ񶯵ĵ㡻Լڸƽתĵ㡻ʶͶӰһԪߵӰʱڶԪԲʶͶӰʱָܷߴԪĽṹͨױ

ʵߣٴߵƽ棬ϵ˼ߵĴԪǱ־ͱüˣ˵ԽˣԽ⡻ɡһֵУȻ˼һⲢֻ⣬Ҫ̽һ֮Ľṹ
˵
Ҫ۾Ȼ۾
׶ȱ˵ָָԼ۾

ҪṹҪۡ


33صĻ

ô¸⡣׶˵

33
nʾһc<n>

n   0    1      2        3      4        5   
c<n>1(1+<3>i)/2(13<3>i)/2  1  (1+<3>i)/2 (13<3>i)/2 

ʲôУ˵
ţ㻹֪𣿡
ʱңֱʵرֳľȣĲֵָ֪𣿡ĸо
ȻľҲҸеҽĸһԣصѧĻϡ
˵1(1+<3>i)/2(1<3>i)/23᲻ϵظ֡ûĴ𰸿𣿡һСעıһ߻ش
Ĵ𰸡û⡢ûṹûץסʡشøɾ䡣
еıǣ
ʾ1(1+<3>i)/2(1<3>i)/2 3ʲô壬Ȼֻ㲻֪3Ļʹһеķʼ׶˵
һеķǾԿȲаɡҿʼڱʼǱд¡
c<n>·d<n>

d<n>c<n1>c<n>(n012)

c<0>c<1>c<2>c<3>c<4>c<5>
  |  /  |  /  |  /  |  /  |  /
d<0>d<1>d<2>d<3>d<4>

c<1>c<0> c<2>c<1> c<3>c<2>˳<dn>

n  0         1    2          3
dn(3+<3>i)/2<3>i(3+<3>i)/23+<3>i)/2   

ţȫ
֪𣿡׶ʵʱ׶ȷ᲻˼غģǽ֮ǰĻͻһǰ壬ǻʱͲż
áʵش
еĹ߾ֻеȲ𣿡Ц˵
˲ʣˡһش
ǾͿԿɡ
ǡǡλ˼e<n>(c<n1>)/(c<n>)e<n>Уc<n>0Բõĳ0ǡ
n        0           1        2
e<n> (1+<3>i)/2(1+<3>i)/2(1+<3>i)/2  

มȫ ˷ܵȭͷ
ھʲô
Ϊñһ
԰ɡc<n>ǵһΪ1Ϊ(1+<3>i)/2ĵȱСʵ1(1+<3>i)/2(13<3>i)/233η1มҲ˵3㡭

x<>1

ηʽ
x<>1
ԣΪx<>1ηʽĸ3֪ʽĽ𣿡׶ʡ
ţӦ֪֪x1ĻͿ԰(x1)ʽֽ⡣˵

 x<>1ķʽ
x<>101ұ߳Ϊ0
(x1)(x<>x1)0ʽֽ⡣

Ȼأ׶˵
Ȼٽx<>x10ηʽax<>bxc0Ľⷨʽxb<(b<ƽ>4ac>)ͿԽˡһ˵һ߼㡣
x1(1+<3>i)/2(13<3>i)/2

ҵ˵׶ȵ˵ͷ
ûڽ(1+<3>i)/2Ϊء

أ(1+<3>i)/2

<ƽ>(13<3>i)/2
<ƽ>((13<3>i)/2)<ƽ>
 (13<3>i)<ƽ>/2<ƽ>
 ((1)<ƽ>2<3>i+(<3>i)<ƽ>)/4
 (12<3>i3)/4
 (22<3>i)/4
 (1<3>i)/2

1ϦصĻͻС׶ڱʼǱϽȥд

1أ<ƽ><><4η><5η>

Ϊ31дģ

1أ<ƽ>1أ<ƽ>

˵1أ<ƽ>1أ<ƽ>͵c<n>(1أ<ƽ>)ڸƽϣ㡣
׶ƺܸˡ

[ͼƽֱϵĵ(10)(01)(10)(01)ʱ˳ͷӣ(10)(01)(01)(10)(10)(01)(01)(10)ռͷһԲΡֱ֮ͷ˳(10)(1/2<3>i/2)(1/2<3>i/2)]

มΰ
뵽ԲǺȻģԲѰҲظԴҲǺȻģֻʵߵ˻ԡ񶯡֣Ըƽ۲˻ע⵽תע⵽صĽṹ԰ɣ
׶ȵϷϺ쳱Ҳűࡣ

33

c<n><nη>(n0123)

Ǧأ(1+<3>i)/2

ΪֹѾ̸4ƽֵΣ3ƽֵΣҪǹ廯nƽֵnˣĹ̡

(ֳ֮Ī)

(cos ȣisin )<nη>cos nȣisin n

šcos ȣisin ȵnηɸcos nȣisin nȡͼεĹ۵ĻǡڵλԲϣתnλnȵתӦܿʽıеλԲϵĵתŶԡ׶ָָңʼȦȦ
n2ʱ򣬾͵õǹʽ

(cos ȣisin )<nη>cos nȣisin nȡ
(cos ȣisin )<ƽ>cos 2ȣisin 2ȡʹn2
 cos <ƽ>ȣi2cos sin ȣsin <ƽ>ȣcos 2ȣisin 2ȡչ
(cos <ƽ>ȣsin <ƽ>)i2cos sin ȣcos 2ȣisin 2ȡ

֮ٽߵʵ鲿õȺᡣ

(cos<ƽ>ȣsin<ƽ>)i2cossinȣcos2ȣisin2
		ʵ	            鲿         ʵ     鲿

͵õǹʽ׶˵

cos<ƽ>ȣsin<ƽ>ȣcos 2 ʵ
2sincosȣsin2ȡ           鲿

㲻ڦȵתг𣿼ȻǳΣͽתĵ㻯ͼΡǺ븴һܲǸ𣿡
ұ׶ȵѹȫ˵
ӷ˵λԲ3ƽֵ<>1ʼŷ2/3ķǡƽϵΣЦزתҲڸƽϿ1أ<ƽ>赸
׶һѻ˵꣬Ȼ¶Цݡ
㿴صĻ𣿡

4¡쳲ɺ


֪У
ʹķ
Ŭ˹ϣ(̷ֳ)ѧ[21]


41ͼ

Ǹ߶죬ڷѧͼҽѧáٵѧĿǰչһ򵥵ʽ

(ab)(ab)(ab)a(ab)b
aabaabbb
a<ƽ>b<ƽ>

ҽ(ab)(ab)չa<ƽ>b<ƽ>˵԰ѡĻƽĲ⹫ʽǻشҶˣѧĽѧоԭ֪ʶˡ仰
׶ȴʱ߽ͼңֱ߽ǣͻȻ߿ٵٵӣ޴ͼ˶ǿٵٻæվ׶ȵֱ뿪ͼңҾվĿ͵ٵٳȥ

׶ûһ£ĿʼǱȻҵҪ£ȵº׶ʣ

ʽ

һشΪѧԽ
׶ȺһԶǦߣڱʼǱϲͣдţȻ˵Ѱҹɰɡ


411Ѱҹ

 Ѱҹɰɡ(1x)(1x)չ(ab)(ab)״

(1x)(1x)(1x)1{1x)x
(1x)(xx<ƽ>)
1(xx)x<ƽ>
1x<ƽ>

ʽ(1x)(1x)е(1x)(1xx<ƽ>)롣

(1xx<ƽ>)(1x)(1xx<ƽ>)1(1xx<ƽ>)x
(1xx<ƽ>)(xx<ƽ>x<>)
1(xx)(x<ƽ>x<ƽ>)x<>
1x<>

ɺԣмȥˣֻʣͳñĻ⣬(1xx2x3)(1x)ʾԵֻʣ

1xx<ƽ>x<>
             1x
--------------------------------------------
xx<ƽ>x<>x<4η>
 1xx<ƽ>x<>
--------------------------------------------
 1x<4η>

廯nΪ0ϵ֮ʾ

 (1)(1x)1x1
(1x)(1x)1x<ƽ>
 (1xx<ƽ>)(1x)1x<>
 (1xx<ƽ>x<>)(1x)1x<4η>
					.
					.
					.
(1xx<ƽ>x<>x<4η>x<nη>)(1x)1x<nη>1

򡡡򡡡

ԭˣⲢûȤǺܳչ廯Ҹոձӵĵٵڵ״׶ȴ˵Ϊֹͨ󣬾ͼдȥ


412ȱеĺ

ΪֹͨôôأղŵʽдһΡ

(1xx<ƽ>x<>x<4η>x<nη>)(1x)1x<nη>1

ｫͬ1x0ܵԼ1x0

1xx<ƽ>x<>x<4η>x<nη>  

ղΪֹǹڡĹʽǡ͵Ĺʽ:ʵϣǵȱе͹ʽ˵øһ㣬ǵ0Ϊ1ΪxĵȱУҲ1xx<ƽ>x<>x<nη>дӵ0nĺ͡
ôҪôչأ

򡡡򡡡

ǻش𡺻Ȼ뵽ȱеɡǵnΪֹ޺ͣ޵ĺ͡ǰ׶΢ЦŻشҡ


413

ǰ˼ɡ
1xx<ƽ>x<>Ϊȱвֺ͵ļޡ

1xx<ƽ>x<>x<4η>x<nη> (1x<n+1η>)/(1x)

xľֵС1Ҳ|x|1ʱʹn޵Ļx<nη>10Ҳʽ

1xx<ƽ>x<> 1/(1x)

͵õĹʽ|x|1Ϊnʱx<nη>10Ҫ

ȱе(ȱȼĹʽ)

1xx<ƽ>x<> 1/(1x)

ǵ0Ϊ1Ϊxĵȱȼ|x|1

㲻úȤеĺͣ޸һ޷ȫдԵģұȴֻһĺ;Ȼһ־ͿˣⲻǼൽ𣿡

򡡡򡡡

ѾͼֻʣҺ׶ȣ׶ƺˣһشͽ˵ͼɺǰɡ


414ɺ

ͼɺǰɡ
ԺʡȽղŵȱȼxĺ˼
1xx<ƽ>x<>

ΪҪô˺nϵԣԽϵд

1x<0η>1x<1η>1x<ƽ>1x<>

ϵγ1111УĶӦǡ

С              
1111   1xx<ƽ>x<>

Ҳ˵С11111xx<ƽ>x<>Եһģ1xx<ƽ>x<>1/(1x)Խû

С           
1111      1/(1x)

뺯ĶӦ¹廯

С                            
a<0>a<1>a<2>a<3> a<0>a<1>xa<2>x<ƽ>a<3>x<>

жӦĺΪɺǽɢϳһɺxݵ޺ͣҲǱΪݼ

򡡡򡡡

˵׶ذ˼ƵرֳĬ˫ۡغ
Ϊ˲ֻһԾؿţ촽õߡӦĺ۾ȱеԼɺ
׶ſ˫ۡ
˼Ӧɺɣ׶ͻ˵ţɺıչʽ˱չʽҲжӦ
΢һ¡˵˵ţҲ͸¶ܱ֪ıسһСңŵϵ㡣
ڿʼҪȼرܡ׶ϸ
Ϊ˲©ЩܵĻ䣬ȣ
ץסУҪֱץסǺѵģʱҪӡеĹȡɺĹȡ֮ٻصеĹȡץС

ǱУʱ䡣

ͻǶһһר۵ȫûע⵽ͼԱʦվ

ɺĶӦ

С                            
a<0>a<1>a<2>a<3> a<0>a<1>xa<2>x<ƽ>a<3>x<>

42ץס쳲

ϵƶϺеĻ⣬ץסУʲô˼ʲôҵʣ׶ַ۾˵ǰʼ⡣



421쳲

ǰ΢쿪˵㡣ȼץסСǡʹɺеһ˼
еͼɣжӦɺɺչʽȻ󽫱չʽݼչȡеһҲ˵ɺԷеһ

ʹɺеһеͼ

Сɺ

еһɺıչʽ

ôԵ͵쳲˼֪쳲аɡ

{0112358}

ǽٽӵӶõһС

01011112123235358

ȻҲд1ʼھʹ0ʼ
쳲еһΪF<n> F<0>0F<1>1n2ʱF<n>Fn2F<n1>Ҳ˵F<n>Ϊһֵƹʽ

쳲еĶ(ƹʽ)

F<n> 0     (n0)
F<n> 1     (n1)
F<n> Fn2F<n1>(n2)

ܽ쳲СڵӶõһԳչ֣ͬF<0>F<1>F<2>һ쳲пüѭ򽥽𰸣ǲ޷ֳF<n>ΪnıչʽҲF<n>ֱʹnʽӣ˵δ׽״̬
ǡ쳲еĵ1000ΪΣʱͱF<0>F<1>F<2>F<1>F<2>F<3>㵽F<998>F<999>óF<1000>Ҫõƹʽ쳲оͱ뾭n1εļӷ̫鷳ˣϣF<n>ʾnıչʽnıչʽǡ֪ļ޵Ĵ϶ɵĹʽ
ϣF<1>ʾnıչʽץס쳲С

41
쳲еһF<n>ʾnıչʽ


422쳲еɺ

ô쳲жӦɺF(x)Ҳ˵Ӧϵ£

              ɺ
F<0>F<1>F<2>F<3>     F(x)

F(x)ϵx<n>ΪF<n>ԾдʽǾ͵ɺĹˡ

F(x)F<0>x<0η>F<1>x<1η>F<2>x<ƽ>F<3>x<>F<4>x<4η>
0x<0η>1x<1η>1x<ƽ>2x<>3x<4η>
xx<ƽ>2x<>3x<4η>

Ҫ麯F(x)ʣںF(x)ϵFn쳲УһƺͿԿںF(x)Ȥʡ
쳲еʲôȻǵƹʽF<n>F<n2>F<n1>ҪúʽF<n2>F<n1>F<n>ЩϵF(x)µǳ

F(x)Fn22x<n2η>F<n1>x<n1η>F<n>x<nη>

Ȼ뽫Fn2F<n1>ӣxĴͬ޷ӣ׸ô죿

򡡡򡡡

׶ȿҡţȷxĴηͬĻ޷ӣͬ޷ϣɺĶӦҪxĴɣɺҪζӦĻôȤ鷢׶˵ܼ򵥡仰


423չʽ

ܼ򵥡
xĴηͬĻͽǵĲֲxͺˣ˷ָϾͱɼӷҲνָ

x<n2η>x<ƽ>x<n2+2η>x<nη>

F<n2>x<n2η>x<ƽ>ͻFn2x<nη>ʽȫϳx<nη>Ϊ˷ϣｫ1Ϊx<0η>
F<n2>x<n2η>x<2>F<n2>x<nη>
F<n1>x<n1η>x<1>F<n1>x<nη>
F<n0>x<n0η>x<0>F<n0>x<nη>

ܶԺF(x)ʹ쳲еĵƹʽˣֱ۲F(x)x<ƽ>x<1η>x<0η>ʽӰɡ

ʽAF(x)x<ƽ>                  F<0>x<ƽ>F<1>x<>F<2>x<4η>
ʽBF(x)x<1η>        F<0>x<1η>F<1>x<ƽ>F<2>x<>F<3>x<4η>
ʽCF(x)x<0η>F<0>x<0η>F<1>x<1η>F<2>x<ƽ>F<3>x<>F<4>x<4η>

ηˣչʽABCнļ㣬һͬηϵͿʹF(x)ĵƹʽ



ʽAʽBʽC

ʱ򣬽´

()F(x)x<ƽ>F(x)x<1η>F(x)x<0η>
F(x)(x<ƽ>x<1η>x<0η>)

Ȼұ¡

(ұ)F<0>x<1>F<0>x<0η>F<1>x<1η>
(F<0>F<1>F<2>)x<ƽ>
(F<1>F<2>F<3>)x<>
(F<2>F<3>F<4>)x<4η>

(F<n2>F<n1>F<n>)x<nη>


ұһʼF<0>x<1η>F<0>x<0η>F<1>x<1η>ᱻȫȥΪ쳲еĵƹʽF<0>x<1η>F<0>x<0η>F<1>x<1η>0ˬؽ
ѾҪx<0η>x<1η>鷳дˣص1xȻF<0>0F<1>1룬õʽӡ

F(x)(x<ƽ>x1)x

ͬx<ƽ>x1͵õF(x)ıչʽF(x)ģ

F(x)x/(1xx<ƽ>)

쳲еɺԱ˵ıչʽ죬Ͼʽпǰ쳲Уܼɡ

    0112358   x/(1xx<ƽ>)

쳲еɺF(x)ıչʽ

F(x)x/(1xx<ƽ>)



424ʾ

˼쳲еɺF(x)F(x)ıչʽxʾĻnεϵӦþͻFn
˵һĿǽʽx֡

 x/(1xx<ƽ>)

֮ǰʽʽx

 1/(1x)1xx<ƽ>x<>

ԣǷܽx/(1xx<ƽ>)1/(1x)ƵʽأԵĻǾܴɺĹȻصеĹˣ쳲еһûа취أ

򡡡򡡡

׶͵͵ҿˣɺF(x)ʽʾĻܵõ쳲еһɣרעɺʽϣ뽫Ĺ쿴͸

F(x)x/(1xx<ƽ>)

ĸ1xx<ƽ>Ϊʽ֮Ƚ1xx<ƽ>ʽֽ⿴ҿʼڱʼǱϼ㣬׶ֻǿҶʡ
δ֪Ķrsɽ1xx<ƽ>ʽֽ⡣

1xx<ƽ>(1rx)(1sx)

ʽֽ󣬾Ϳ÷ͨʱպʹĸΪ1xx<ƽ>

    1/(1+rx)+1/(1sx)=ĳ/(1rx)(1sx)
		     =ĳ/(1xx<ƽ>)

Ϊô˼ʽx/(1xx<ƽ>)ͱrsż㿴

 
1/(1+rx)+1/(1sx)(1sx)/(1rx)(1sx)+(1rx)/(1rx)(1sx)
		     (2(r+s)x)/(1(r+s)x+rsx<ƽ>)

ţúѡrsĻĸ1(rs)xrsx<ƽ>ƺԱ1xx<ƽ>Ƿ2(rs)xĲ޷xԭǳ2޷ȻܿϧȻǲУź
ҵ૵ͬʱ׶ȴ˵Ļ֮ûˡ仰



425

Ļ֮ûˡ
ҲҲ˵RSrs4δ֪γһʽ

 R/(1rx)+S/(1sx)

Ȼ㡣

 
R/(1rx)+S/(1sx)R(1sx)/(1rx)(1sx)+S(1rx)/(1rx)(1sx)
		     ((R+S)(rR+sS)x)/(1(r+s)x+rsx<ƽ>)

ΪʽҪRSrs4

((R+S)(rR+sS)x)/(1(r+s)x+rsx<ƽ>)x/(1xx<ƽ>)

Ƚ߽ṹֻᷢҪµʽɡ

RS0
rSsR1
rs1
rs1

ĸδ֪ĸʽӣŽʽɣֻʣ¶ˡ

ȽRSrsʾ

R1/(rs)S1/(rs)

ͿõF(x)ʽʾķˣΪ֮rsȥ

F(x)x/(1xx<ƽ>)
x/(1rx)(1sx)
R/(1rx)+S/(1sx)

R1/(rs)S1/(rs)
(1/(rs))(1/(1rx))(1/(rs))(1/(1sx))
(1/(rs))(1/(1rx)1/(1sx))

Ȼ 1rxr<ƽ>x<ƽ>r<>x<> 1sxs<ƽ>x<ƽ>s<>x<>
(1/(rs))((1rxr<ƽ>x<ƽ>r<>x<>)(1/(rs))(1sxs<ƽ>x<ƽ>s<>x<>))
(1/(rs))((rs)x(r<ƽ>s<ƽ>)x<ƽ>(r<>s<>)x<ƽ>)
((rs)/(rs))x+((r<ƽ>s<ƽ>)/(rs))x<ƽ>((r<>s<>)/(rs))x<>

֮ͱ

F(x)0((rs)/(rs))x+((r<ƽ>s<ƽ>)/(rs))x<ƽ>((r<>s<>)/(rs))x<>
F0      F1            F2                              F3

ܽ쳲еһrsʾ

F<n>(r<nη>s<nη>)/(rs)

֮ʣrsˣrsʽ£

 rs1
     rs1

ҪһʽķʽҲԣǺΪ1һΪ1rsڷx2(rs)rs0Ľ⣬ҲνġηʽĽϵĹϵʽֽ⣺

x<ƽ>(rs)rs(xr)(xs)

Ҳ˵rs1rs1ԵõxrsΪ·Ľ⡣

x<ƽ>(rs)xrsx<ƽ>x1
         0

öηʽĹʽԵõ½

 x(1<5>)/2

ٶrs򡭡

r(1<5>)/2
s(1<5>)/2

rs<5>ԡ

 (r<nη>s<nη>)/(rs)=(1/<5>)(((1<5>)/2)<nη>-((1<5>)/2)<nη>)

Եõ쳲һF<n>¡


F<n>(1/<5>)(((1<5>)/2)<nη>-((1<5>)/2)<nη>)

ãͽˡ

41(쳲еһ)

F<n>(1/<5>)(((1<5>)/2)<nη>-((1<5>)/2)<nη>)


43ع

һ޷ܣ𣿱Ͼ쳲ȫҲΪһ<5>
ϸ׶һѾĿȺ꣬ҵǻش𣺡ҪԿ𣿡
ôn01234Ϊ㡣

F<0>(1/<5>)(((1<5>)/2)<0η>-((1<5>)/2)<0η>)0/<5>0
F<1>(1/<5>)(((1<5>)/2)<1η>-((1<5>)/2)<1η>)<5>/<5>1
F<2>(1/<5>)(((1<5>)/2)<ƽ>-((1<5>)/2)<ƽ>)4<5>/4<5>1
F<3>(1/<5>)(((1<5>)/2)<>-((1<5>)/2)<>)16<5>/8<5>2
F<4>(1/<5>)(((1<5>)/2)<4η>-((1<5>)/2)<4η>)48<5>/16<5>3


01123ȷʵ쳲Сมԭˣʵʼʱ򣬷Ӹĸͬʱ<5>
ţһߺſȣһ߻ع˽̣쳲еһ(ҲǶnıչʽ)Ϊµ˳ǰ

(1)˼쳲еFnϵйصɺF(x)
(2)F(x)ıչʽ(Ϊxıչʽ)ʹ쳲еĵƹʽ
(3)F(x)ıչʽʽʾʱϵxnΪ쳲еһ

Ҳ˵ʹϵͬĺɺһһץСˣԭˡ;

쳲еһеͼ
쳲С   ɺF(x)
 
쳲һɺF(x)ıչʽ

׶ȿʼ˵ɺǲеЧԭڣ֪ĺɺĹ﷢ЧãںļҲܻеоС
ұ׶˵ߵı顣ڼʱ򣬱͵˳ǲ̫׶ȡû
˵ĻȷʵãξͲҪƽˣʹɺֵĶܣѧɷ֤óһͿˡ
׶ˬػش

չʽʱ
ʹɺҪֶΣ
ΪҪʾҳʽķ
ߵɡThe Art of Computer Programming[22]



5¡ƽ뼸ƽĹϵ


һдϲã
ڿʧıԵǻ
˹ء᥿ޥå`ࡻ[5]


51ڡѧ֡

졣
ѧҼæУ԰Уһ߿첽ǰһߴӿڴóֽٶȶһֻֽдһ֡

ѧڡѧ֡㡣ٵǉ

ݵĴֳΪѧ֡ͬʱٵѾڵˡ
ٵһҾͼæͷ
ǷǳǸǸ
õǸҲŶԡ֮Ƚȥɣ̫ˡ
ѧ֡ǸʵĿռ䣬и⣬ҲΣ죬˻Ǻܶ࣬¥ϵŵĻңϰѡ
Զһ޿ȣ˸ʵλ£ٵҵĶ档
ٵǸһ꼶ͬгѧãȻô˵ҹʱʶ
ҡһʲôû˵ֱӻȥˣǷǳǸٵصͷ
ҲӦõǸкܶԭ򡭡
ٵٺܽŵؿң΢ЦӦٵٵ۾ִԲȴǳС˵ĸоŪҵɵβƺʮʺ
ǡǸѧѧѧǲǺλѧڽ

ڽ𣿡
׶ȰûУǲڽ
׶ȡ
ҵиֳģҲȷ˴𰸣ţǲûڽ
ȷʵΪҺƤѧԱߣŻᡭ¡ٵٱעҵɫ˵ȻǸ磬ѧ鷳Ļ֮󻹿ѧ𣿡
ţ԰绹ʵ⣬ʱң֮ǰһžΪϣܹ֮㣿
ҿֽٵٵͷ
صؽǶԲȻֱӵͼȥ
һٱ׶һεĻȷʵܲ˰ɡ
ǡǸѧҵĻλѧ梻ᡭýҵ𣿡
ٵֵҲ¶Ц
׶ȰáýӰɣô˵ءţ׶Ǳ߾˵ˡ
ٵҵĻ¶΢Ц



52

ҴӺܾǰиʣһֱûѧᵽ(ab)(ab)a<ƽ>b<ƽ>ʽҿ˲ο飬д(xy)(xy)x<ƽ>y<ƽ>
ţͬĹʽʲôط⣿
ǰʹabʹxyı߱ȽϺأ
ԭˡ
ңÿѧʽʱΪʲôдأʣ֮ͺѼȥˡҪʦȴֲ֪ǲǸʡ֮ԽԽѧˡ
ԽԽ᣿
Ҳʲô¶Ҫȱ˶໨һʱ䣬᲻ϵʣõĴȴһЩѶĶԾͻὥؿʼᷳ
ԭˡ
ҴŲʺѧѧɣѧܺõѣҲҵڷʲô˵Щùܡ˵ùʱֱ˵ҪúעⰡǸעĵطѾȫˡ
ʷʺѧѧ˵
ӢĻٵټ˵šֲԲֵ䣬ѶĹþǣķȻ鷳ǺһǾͺˣֻҪжٶ
᲻̫һ㣿Ȼô˵ȻΪ˲Ļҵͷ
ѧͲˣʱǲʱȫû취ûνԡ
šҲ㵽;˳״һش
ѧ˵ĲԲ𡣴Ӹոտʼһֱ¿ˮ¿ˮ¿ˮҲ¿ˮǶùúùġ
ٵһ˵ùһ߽ȭͷս
ҺܽУҲϣ¼صĹ벻ܳĸǰѧǱҪġԲŻҪŬù
ٵһصͷ
ѧͨôùأ
ʱǽ⣬ʱֻǵддʽѣƩ˵ˣǾһùɡ
áõġ


53ʽ

ٵ˵һʧˡ󣬾ƶԱߵλһűʼǱ΢΢ζǺ׶ȲһζȻġ
ǾͿʼɣrʵrĶηr<ƽ>˽أ뿴
ҵʣٵ˼롣
Ϊr<ƽ>ƽԻ0԰ɣ
ǵģȻǡ0ǡڵ0Ŷԡ0ĻͲ0ˡ
ҮrĻr<ƽ>Ҳ㡣õģr<ƽ>ڵ0
ٵƵصͷ
Ҳ˵rκʵĲʽ԰ɣ

r<ƽ>0

ףǰrʵĻr<ƽ>ʹڵ0ˡ
ʵrǸƽ֮󶼻ڵ0r<ƽ>0ᵽrΪʵʱҪرעҪʣȺųʱr0״
ʡȻİɡ
ǵģȻȻĵطʼǺõĵһ
õġ
ʽr<ƽ>0κʵrԳκʵԳĲʽΪԲʽ
Բʽ
ӡκֹ۵ĻԲʽʽƣֻھԲʽǲʽʽǵʽ
ԭˡ

ټǰɡabΪʵʱʽܹɣ

(ab)<ƽ>0

šûabΪʵΪʵƽ0ϡһ£ղr<ƽ>0ʱùrĸΪʲôξabأÿζΪʲôҪڵʱʦͼ˵ȥˡ
ûϵղõrʵ(rea1 number)ĵһĸҲxΪʵһʹxǡwΪʵһʹwͨ˵ʱʹabcʱͻxyzĻԣȻˣдnΪʵʱǻһϾnͨڱʾȻţΪֹ𣿡
ԣǳ˼ѧĻÿζʹõ֡(ab)<ƽ>0Ѿˡ
ٵ¶Цݣҫ˵أǸḻŮһҪȫŻȥһҲܲ
ôҪǰźã
õʣٵô۾ش
ǡﰡ
ʲôԡ⡭

(ab)<ƽ>0

֮Ҫ˽ʲôʽκζԣ˵˵ɣҪдд
ҽԶǦʽٵ١
õġҾչˡ

(ab)<ƽ>(ab)(ab)
(ab)a(ab)b
aabaabbb
a<ƽ>2abb<ƽ>

𣿡
ţ԰뿴ʽӡ

(ab)<ƽ>0(ab)2a<ƽ>2abb<ƽ>

Ϊ(ab)2Ϊ0չĽҲ0ϰɣٵ١
ʽĵٵͻȻ̧ͷգգ۾֮Цţƺܸˡ
ǵģû֮Ҫôأ
ţڿʼűʽӣ罫2abұ߿֮2abͱ2abˡ

a<ƽ>b<ƽ>2ab

ǣҶ
ͬ2Ȼͻ

 (a<ƽ>b<ƽ>)/2ab

š
ʽʲôأ
ʲôأ
ߣᷢ(a<ƽ>b<ƽ>)/2a<ƽ>b<ƽ>ƽ
ԭˡǽa<ƽ>b<ƽ>ӣٳ2
ţʽӵдa<ƽ>b<ƽ>ҪұߵabҲһa<ƽ>b<ƽ>ʾ

ûΪʲôһҪĹֻʱһѡ
õġ
һ΢е죬ҪעมΪ˽aba2b2ʾҪγʽΪʽȻ𣿡

ab<a<ƽ>b<ƽ>>()

ƽ֮Ϳõţõ֮󣬻ظԭӡԣӦǱȻ
ԣƽȥ֮ظԭ״ֻ0ϵabпǸûĻʽ
ѽˡ
ǰa2b2֪ˡab2(2)4ұȴ<a<ƽ>b<ƽ>><2<ƽ>(-2)<ƽ>><16>4
ٵһȷдļʽһߵͷ
ξͼab0Ļĵʽͻˡ

ab<a<ƽ>b<ƽ>>ab0

Ȼ󽫸ղŵĲʽ(a<ƽ>b<ƽ>)/2abдɡ
 (a<ƽ>b<ƽ>)/2<a<ƽ>b<ƽ>>ab0

õġȻٵش𣬲Ծרĵ˼
һ£ѧֵֹġab0ǱҪһǲab0ʱʽҲ𣿾ӣa2b2ʱ߻ֱ¡

ߣ(a<ƽ>b<ƽ>)/2
(2<ƽ>(-2)<ƽ>)/2
4
ұߣ<a<ƽ>b<ƽ>>
<2<ƽ>(-2)<ƽ>>
<16>
4


ߡұ߻รѧ
ܷأٵ٣ȷʵab0ҲԣҪôȡأ
ٵһ£ҡҡͷ
֪
Ҫȡab0ͱ֤ʹab0ʽҲ
ab0ʱabһģһǸģȼٶa0b0ˣcbc˼Ϊb0c0(a<ƽ>b<ƽ>)/2abκʵԶacҲʽ

 (a<ƽ>c<ƽ>)/2ac

Ȼҷֿ

ߣ(a<ƽ>c<ƽ>)/2
(a<ƽ>(-b)<ƽ>)/2Ϊcb
(a<ƽ>b<ƽ>)/2
ұߣac
<a<ƽ>c<ƽ>>Ϊac0
<a<ƽ>(-b)<ƽ>>Ϊcb
a<ƽ>b<ƽ>

ʽӳ

(a<ƽ>b<ƽ>)/2a<ƽ>b<ƽ>a0b0

Ǽ衺aΪbΪ˵aΪbΪҲһ֤ʵabĲʽ

(a<ƽ>b<ƽ>)/2a<ƽ>b<ƽ>abΪʵ

ٵٿдڱʼǱϵʽ˼Ȼһʱ䣬̧ͷ
ԭˣҶˡһ㣬ν⡻ʲô˼
ν⡻ǡκһǡһ˼൱ӢĵanyҲáеġʾķӢĵĻǡfor all
֪ˣʵָһʵ˼ɡ
Ҽ˵ȥ
ｫߵʽa<ƽ>b2ʾa<ƽ>b<ƽ>Աֱ֣a<ƽ>Ϊxb<ƽ>Ϊyʽ¡

xa<ƽ>yb<ƽ>

xyƽԶ0ϡҲ˵x0y0ԸղŵĲʽͿʽӱʾӦúˣпʽӰɣ

(x+y)/2<xy>x0y0

֪šƽ뼸ƽĹϵ
ţʽǡӳ2ƽ(x+y)/2ұǡ˿šļƽ<xy>νƽ뼸ƽĹϵָƽһСڼƽ
ǵģȻr<ƽ>0ʼһֱݵʽˡٵٸп˵롣
˵ɡʽͺбǵӡײ׽ĸоǼ糣ʽ仯ϰ뷨ͻὥԽԽ
มԭʽԼġ
˵Լģ˵ǴʽﵼģʵѧαؼҵҲʽĿԺŶע⿴ݵڿαϰС
һעģԭᵽʽͻСϿĸоء
ʽƵһʼǵķѼҪԼּ㣬Ȼ⣬ǺҪģûǵĻͨûõġ
ԭˡ
˵֪ƽ뼸ƽĹϵУʲôʱȺŻҲ˵ʽӳʱxyʲôĹϵأ

     (x+y)/2<xy>
ףxy0ʱ𣿡
ԡӦ˵ȫȷ
ףǵx0y0Ļߺұ߶0
뷨ǶԵģǲһҪxy0ɣֻҪxyͺˡ
ףx3y3ȷϿˡ(x+y)/2(3+3)/23ұ<xy><33>Ү
ţʵǺҪģԽʯ
Կx2y2ʱأ(x+y)/2((-2)+(-2))/2-2ұ<xy><(-2)(-2)>2ףˣ
ٵ٣x0y0ม
ѽûûȫ©һֱ붫ؾˡ
ٵͷԼͷ
ٵ٣ֻҪע⵽Ǵ(ab)<ƽ>0ʽʼ֪Ⱥųʱab(Ҳxy)ʱ򡣡

ƽ뼸ƽĹϵ

(x+y)/2<xy>

ҽx0y0xyʱȺų


54һ

ڽʽӣȻһֱظе㷳ˣҪ֤ƽ뼸ƽĹϵʵֻҪ(<x><y>)<ƽ>0չͿˣǼǵx0y0

(<x><y>)<ƽ><x><ƽ>2<x><y><y><ƽ>
 x2<x><y>y
 x2<xy>yΪx0y0
 0Ϊ(<x><y>)<ƽ>0

Ҳǻ

x2<xy>y0

Ȼ2<xy>ͬ2ͻʽӡ

(x+y)/2<xy>x0y0

ףx0y0ģ
Ϊֻʵ<>֮xy0ϡ
<>֮б0Сأ
0СĻͱˡ
ԭˡ

ǣ΢һƽ뼸ƽĹϵɣղдûбֳƽ͡ƽԽࡣ
ԡࣿ
ԣҪƽķűΡxyxy׵дţ2ó ʾƽ η֡ĻʽͻҲƽ뼸ƽĹϵรд׵ؿߵԡ

(xy)(1/2)(xy)<(1/2)η>(x0y0)

ٵ̾֡
ѧһ⣬ƽ<>ɣ(1/2)ηʲôأ
νȡƽָ(1/2)η˼Ȼ˵ ηܻһǶ塭һʼָķ˼Ļͺܺ
(1/2)ηܺ
Ҽ˵һ¹x0xƽx<(1/2)η>°ɣ˼һʲô(x<>)<ƽ>
(x<>)<ƽ>Ϊ(xxx)<ƽ>ܹ6ηӦ(x<>)<ƽ>x<6η>
ûͨʽһηĴηָĳ˷㡣

(x<aη>)<bη>x<abη>

˽⡣
ԭʽӣaҪʲôȽϺأ

(x<aη>)<ƽ>x<1η>

Ϊָĳ˷a1Ծa(1/2)
ţȻ뷨˵ϸ(x<aη>)<ƽ>x<1η>Ϊx<1η>xʽǡx<aη>ƽxʹx<aη>ǡ
ƽĻxΪ0ϵɡǾ<x>ۡ
ɣһ(1/2)ηƽܺɣ

ƽ(1/2)η

x<(1/2)η><x>(x0)

Ȼܲ˼飬ȷʵоܺ
ˣܲܽƽ뼸ƽĹϵ廯֤ʽ˵Ȥ

(x<1>x<2>x<n>)(1/n)(x<1>x<2>x<n>)<(1/n)η>(x<k>0)

ʽʹæǾͻǺͣұǻƽ뼸κĹϵǺ֮Ĳʽ

<k=1n,x<k>>(1/n)ݡ<k=1n,x<k>><(1/n)η>

ѧѧȻȤҺ




55νѧ

΢Ϣһ£ٵٶȻصҶλ̸۹ڶѧĻ⡣
ѧʱ˾ᷳľǲĿ⿪ҲȤڼԼѧģҲõ׽õҲǡѧѧԽʲôֳĻֻ֪ոѧʽε׺ѧ콫ҪѧĶʲôϵϣܿͼȫòʦȴûҿ

ͺСֵͲһȫڷĸоȻֵͲǰǹյķΧȴխ޷֪ԼߵǰһƬڣֻܿйյһСֶѣĺѵĻҲûа취ʵʽı仯ûôѣѾ㲻ѧǼ򵥻ˣƺܼ򵥣ȴ޷壬ûеͼʱ벻
ԭˡ
ٵٵĲҲǡýᷢʲô¡
ѧܹ˵ͬѧͲˣȻҲѧܺõѣǾû취˳̸ÿζᱻ۸˵㲻ҪЩֻҪͺˡʱҾͲЩһЩ˵ȥˡ
Ҿ񱻵ٵٵĻһڣ
ϲѧһͼﲻϵؿʽҲὫϿгֵʽϣԼܣȻһһؼȥѧϰʱһرעԼܲܽЩ֡
ٵٰ˵
ѧУֻṩѧϰزģʦҲֻĿԵ飬ЩҪֻҪϵ˼ԼȤⲢǱĸǿȵģʲôʽĸģֻע⵽ǲǰϲ£ԭǾͲ̫ر顣
ΪѧĳɼðҾͲУȥ须Ƿˡ
ҳͼ飬򿪱ʼǱʽʽΪʲôǵҪ岻ɣ߳Ըı䶨ᷢʲô£ҪĵطԼ˼㱧ԹʦͬѧҲû塣ٵٸո˵ʽʱ롺ΪʲôҪдأԲǻ£ܻʱ䣬ǲ׶ԼֵЭŬ˼ǺҪģҾ飬Ǹĸѣʦûа취شٵԼɻ޷ȫػشʱԼ޷شʱչַŭˣȻͷȥЦǡ
ѧͼҽҵʱҲҾȤȻֻǼ򵥵ʽΣȴȸԾĸоĻҲᵱɲοġʡѧЩиλѧ˵𣿡
λѧ梣
׶ѧ梡
ţô˵Ӧ˵ݸӾɣͼҼʱʱ׶Ȼչۣڼݣ󲿷ֶ˵׶ȺܴӮ˽ıҸ㡣
Ϊѧλѧ梡ڽΪһ𡣡
Ϊͬడ
ͼҲǡ

Ǹѧȫ꼶һɡ
ǣѧ׶ȵһһжڣ˵Ļǵһ
ΪʲôҶôأ
ΪҶϲȥѧϰģ˶Ҳ⣬Һ׶ȶó˶Ȳ׶ȣҺܲóڴǰ˵ϲѧΪϲԲŻȥٵϲĶ𣿡
ϲӢǳϲ
зӢɣҵȥʱ򣬻Ӣĵߣ԰ɣ
ûѧѧ
ΪҲһҵĻͻߣǵļ궼һǳȥ꣬ڷŵλöֻҪһܾ֪Щ顣ֻˣֻԼϲ飬ֻǻʱԼϲϣΪϲʱ䣬κ˶һϣ롢˼Լϲ£νϲһ°ɡҷ·ĳĿرһһеĸ黯Ϊֳ
ѧУ粻խСҲкܶƭСļȻѧУ֮Ҳкܶ󡣵Ҳ˵Ʒ
ѧУﶼûƷ𣿡
˼˵ĳʦ֪ľʦɣȻ˵Ǹֵˣĺܶ࣬ң׶ȶΪʱᱻľʦ⣬ʱҲȤǡ
ٵͷ˼˵ȥҵĻϻһ򿪾ͣ¼С
׷ϲʱͻõֱαһЩϲ£ǹѧڣЩϰ߱ҡǳԼͷ˼ϰߣȥ˽ıʣͲҪǿˣʹҲᶮƹʽҲⲻʽ۱ôΪ۱˵ʲôҪ˼ԼܽΪֹΪ൱Ҫ¡׷ϲ׷ıʣΪ
ͣ˵̫ˣΪԼµ޴Ϊֹɡ
ٵصͷȻ˼֪ʱ¥ϳѵĳϰѾ˲ϰҲˣѧ֡ʼ֡
ѧʱиûõѧá𣿡ٵ÷·Ҫʧ˵š
ף
иûõѧԱߣŵ𣿡
ȫšҲ鷳˵Լ£˷и̸ҷǳˣҲرҪһ˶
ܾúĽѧȻҲŬѧѧǵȼȫһ
ٵҧʳָָס
Ĭ
һٵͻȻ̧ͷ
۱ôֻҪ׷ԼΪԵľͺ˰ɣѧоˣǸϣܰ㣬󡭡ʹֻżҲãҼѧѧ㣡
ٵı
ţ԰
ûʲôϵŰɣҽ񱻵ٵ١СüΣҲش𡺿ԡüˣһûģ˼еҿ˿ʱӡ
ѧҲҪȥͼ𣿡
ţǰ
ҲǸˣȻȥˡ֮ͼһǡڽҵʱ򡣡
԰Ȼ
ֻشһΡ
ʱŮӵٵٵı߹һˣٵǉ
ˣٵҲתǴػش
Ȼ󡭡
ٵְסת˻ڳһˡı飬ƺΪǰֳƽһе˼쵽

ж꼶ĵٵҾ÷ǳؿɰ



6¡׶ȵ


о
оɢ
ߡ
˺[14]


61΢

һû˵ͼʽ
׶Ƚ󣬺ԥԱɢ㣬ڿҵıʼǱ˵
΢֣
ǰһش
׶ֳԱ߿Ҽ㣬û˵κλһֱ˿šҺѼȥ
ôˣ׶˵
ûСڿʲôһش
ļ㡣̵ػش
˵û
Ϊ׶Ȳֻǿ鷳ľ˲ͬԱߵλͻȻӸֵסʽ͸Ҫһ̽
ˣٵٵԼˡ
׶Ǳ߾˵ˡ
ˣ׶ȣ֮ǰǼ¡
һ¡׶˵껰ϲ۾״˫Ὺϣƺǿʼ˼Ȥ飬ҾͲܴˡ
룬ſۣһ߶˵΢֣򵥵˵Ǳ仯ɡһҵıʼǱд

򡡡򡡡

΢֣򵥵˵Ǳ仯ɡ
ֱڵλΪx΢еĵطΪxhh̫Ҳǡ롻

    [ͼһֱߣȡһxȡһxh]

Ȼ˼fı仯ɣӦxĺfΪf(x)ȻӦxhĺfֵΪf(xh)ע⡺Ϊhʱfα仯
ΪʹԱȸĿ0ҲλΪx0ʱfֵf(x0)ǰxhʱfֵͱf(xh)
x0ǰxhı仯á

ǰλáǰǰλá

Ҳ(xh)(x0)hͬأx0ǰxhʱı仯f(xh)f(x0)

[ͼֻƽֱϵϻһf(x)=2<xη>ĺͼߣͼȡһ(x0,f(x0))Ҳȡһ(xh,f(xh)֮ĺΪ(xh)(x0),Ϊf(xh)f(x0)]

ҪxĺfҲǵ˲ı仯xı仯(xh)(x0)Ļfı仯ҲԽó൱ͼббʡ

(fı仯)/(xı仯)(f(xh)f(x0))/((xh)(x0))
Ҫ֪λxı仯h뾡ܵСhϵСh0ļޡ

 lim<h0,(f(xh)f(x0))/((xh)(x0))>

򵥵˵Ǻf΢֣ͼͼĵ(xf(x))бʣԽӽбʴϷf(x)ӵԽ죬Ҳ˵Ǹص仯ܴ

 [ͼֻƽֱϵϻһf(x)=2<xη>ĺͼߣͼȡһ(x0,f(x0))õ]

Ժfġ΢֡дDf仰˵΢D¡

΢DĶ
Df(x)lim<x0,(f(xh)f(x0))/((xh)(x0))>

ҲԶʽӣϾͬĶΣ΢DǴӺĸ߽׺

Df(x)lim<h0,(f(xh)f(x))/h>

ĿǰΪֹǹĻ⣬xɵػڿʼҪɢˣȻɢ磬Ҳһֻȡֵǵǽxƶ롻hfϵı仯Ȼh0ļ޶Ϊ΢֣ô罫΢Ƶɢֻأ

61
Ӧ΢Dɢӡ

62

׶ȵ⣬ֻҪɢҳ硺롻ĸͿ˰ɣһͼһȦȻԵ׶˵롻ɡ˼ûɣ𰸣ʳָش𣺡û

򡡡򡡡

û
˼ɢʱx0ġ롻x1ġҪh0h1˼ɢıʣע⵽۾˳չ

 [ͼ2ֱߣһΪʵߣһΪߡÿֱȡһxȡһxhʵ֮ľΪġ롻֮ľΪɢġ]

x0x1ı仯(x1)(x0)ʱfı仯Ȼf(x1)f(x0)Ҳͬȡߵıһĸԭ1

(f(x1)f(x0))/((x1)(x0))

ɢûбҪȡޣʽӾǡɢ΢֡Ҳǲ֣Ӧ¡

61(ӦĶ)

f(x)(f(x1)f(x0))/((x1)(x0))

Ҳдʽ

f(x)f(x1)f(x)

ĲȷʵʵĦ֡㰡

[ͼֻƽֱϵϻһf(x)=2<xη>ĺͼߣͼȡһ(x0,f(x0))Ҳȡһ(x1,f(x1)֮ĺΪ(x1(x0),Ϊf(x1)f(x0)]

΢ɢĲֲűȽϣΪöԱǿдúܷ

                  ΢֡ ɢĲ
                   Df(x)  f(x)
lim<h0,(f(xh)f(x0))/((xh)(x0))>(f(x1)f(x0))/((x1)(x0))


63΢

׶ƺܸˣÿ˵᲻֪رȥ
Ҫúú˵
׶ȣϴԱߵǸŮ
עӱʼǱƿתͷңһ˲両ɻıֽƻʽϡ
ҹʱѧãȻ󡭡
֪
ף
֮ǰ˵ˡ
ȻעűʼǱ
Ȼʱһѧ
Ҳ֪

˵޷
˵껰ͽԶǦָת

򡡡򡡡


631һκx

˵޷óf(x)Ҫþĺ˼
˵һκf(x)x΢ֻȽϡ
΢֡

Df(x)Dxf(x)xõ
lim<h0,((xh)(x0))/((xh)(x0))>΢DĶõ
lim<h0,1>
1

Ȼǲ֡

f(x)x f(x)xõ
 (x1)(x0)΢DĶõ
 1

΢ͬ1ȷf(x)x΢һġ


632κx<ƽ>

˼κf(x)x<ƽ>΢Ҳһ
΢֡

Df(x)Dx<ƽ> f(x)xõ
lim<h0,((xh)<ƽ>(x0)<ƽ>)/((xh)(x0))>΢DĶõ
lim<h0,(2xhh<ƽ>)/h>
lim<h0,2xh> hԼ
2x

Ȼǲ֡

f(x) x<ƽ>  f(x)xõ
 (((x1)<ƽ>)(x0)<ƽ>)/((x1)(x0))΢DĶõ
 (x1)<ƽ>x<ƽ>
 2x1

x<ƽ>΢2xǲȴ2x1͸ղf(x)xһε΢ֲһ£̫ȤˣҪôǻӦأ

62
Ӧĺx<ƽ>ɢĺ

ҪôΪ׶ȵ˼Ҫβ΢ֻӦǲû뵽ʲôð취ȷûҪش˼֮׶úͻؿʼ˵

򡡡򡡡

ʵԭ޷ɢx<ƽ>Ӧx<ƽ>Ҫɢx<ƽ>˼


f(x)(x0)(x1)

żf(x)(x0)(x1)Ĳ֡

f(x)(x0)(x1)
 ((x1)0)((x1)1)((x0)0)((x0)1)
 (x1)xx(x1)
 2x

㿴ͺ΢һˡ
Ҳ˵x<ƽ>Ӧɢ(x0)(x1)
Ϊx<nη>ݵĶӦµx<nεݽ׳>ݽ׳˼ɣӦ״

 ݡ                ݽ׳
x<ƽ>xx x<2εݽ׳>(x0)(x1)

дøܸ߳ؿӦĹϵ

x<ƽ>  x<2εݽ׳>(x0)(x1)

62(ɢx<ƽ>)

x<2εݽ׳>(x0)(x1)

ʹõĵݽ׳xn¶塣

ݽ׳˵Ķ(nΪ)

x<nεݽ׳>(x0)(x1)((x(n1)))n

˵
x<1εݽ׳>(x0)
x<2εݽ׳>(x0)(x1)
x<3εݽ׳>(x0)(x1)(x2)
x<4εݽ׳>(x0)(x1)(x2)(x3)


633κx<>

˼f(x)x<>
΢֡

Df(x)Dx<>
lim<h0,((xh)<>)(x0)<>))/((xh)(x0))>
lim<h0,((x<>3x<ƽ>h3xh<ƽ>h<>)x<>)/h>
lim<h0,(3x<ƽ>h3xh<ƽ>h<>)/h>
lim<h0,3x<ƽ>3xhh<ƽ>)>
3x<ƽ>Ѳh

ɢУӦx<>x<3εݽ׳>(x0)(x1)(x2)x<>Ĳ֡

f(x)x<3εݽ׳>
(x0)(x1)(x2)
((x1)0)((x1)1)((x1)2)((x0)0)((x0)1)((x0)2)
(x1)(x0)(x1)(x0)(x1)(x2)
((x1)(x2))(x0)(x1)     ȡʽ
3(x0)(x1)
3x<ƽ>

ʹõݽ׳x<nεݽ׳>Ļ΢ֳֶӦ

 x<>       x<3εݽ׳>(x0)(x1)(x2)
Dx<>3x<ƽ>  x<3εݽ׳>3x<ƽ>

廯

x<nη>΢֡            x<nη>Ĳ
Dx<nη>nx<n1η> x<nη>nx<n1εݽ׳>

634ָe<xη>

Ƕ΢Ӷ˲ӣΪ΢꣺ӦԳx˵ݽ׳ˡ
Ҫʼָe<xη>ҲѰɢָ

63
ָe<xη>ɢĺ

ָe<xη>ͬʽʾǶexηeǱΪȻĵ׵ֵΪ2.718281828ȻһҪ֪ʶҪվڸӵ
ָeʲôأ
΢˼һ΢ָֹıʡ
ָe<xη>Ҫʾǡʹ΢Ȼı䡻Ҳ΢e<xη>õĺȻe<xη>Ϊ˶eģֽҲȻ
áʹ΢Ȼı䡻ʣʹ΢D󣬿΢ַʽ֡

De<xη>e<xη>

Ϊֹǹָ
ڿʼҪɢ磬ҪɢָΪE(x)ǾͻϣE(x)СʹȻı䡻ʣʹòӦ󣬿ʽӱ֣ǲַʽ

E(x)E(x)

ԲӦĶ彫չ

E(x1)E(x)E(x)

֮õƹʽ

E(x1)2E(x)

˵ƹʽ0ϵxǺE(x)ʡм1ͻӦ2ܼ򵥵ؽ⿪ƹʽ

E(x1)2E(x)
22E(x1) E(x)2E(x1)
222E(x2)  E(x1)2E(x2)
2222E(x3) E(x2)2E(x3)

2x1E(0)

õʽӡ

E(x1)2x1E(0)

E(0)Ҫôأe<0η>1ԶE(0)1Ǻ׵ģʽɵöӦָe<xη>ĺE(x)¡

E(x)2x

ԾµĶӦϵ

63(ָ)

  硡 ɢ
e<xη> 2<xη>

ɢָ2ĳݣöӦطǳǡ


64ó

˼΢ֲ֮֡λֺͷ֡ɣֻд
˽ıͷġTXTʽ˵Ӧ
1x     1x
  tx<ƽ>/2   t=(x<2εݽ׳>)/2
  t<ƽ>x<>/3   t<2εݽ׳>=(x<3εݽ׳>)/3
  .  
  .  
	.
	t<n1η>x<nη>/n   t<n1εݽ׳>=(x<nεݽ׳>)/n
	t<nη>x<n1η>/n1   t<nεݽ׳>=(x<n1εݽ׳>)/(n1)
ԡҴȫġ<x,0>ԡƴȫġ<x=1,t=0>ֻԵĻµĶԱǿܳġ

 D
ҡ
ԡֵSϣֵSĻԱȾ͸˼ˣɢϣ

򡡡򡡡
һ׶ȵĻ֪ʶΪ̽ɢ磬˵׷ܵĶ壬˵׷ʵĶ̣˼Ӧ΢ֵĲ֣ԴΪ˼Ӧxnxn޷΢ַʽĲַʽҵӦe<xη>2x
粻صó˸еɣֵֿǴأ
׶ȵĻҾȻûа취ġ롻ϣġ

򡡡򡡡

Ȳ
׶ȣղ˵ǸŮ֮Ҳ⡭
ǸŮ
ҵѧá
ǣ
ٵǉ֮Ҳ⡭
ԡҲԱˣ׶ȱдʼǱʡûпҡ
ףǲǣȻ׶ʲôʱҪԱ߶УҪʲôԣֻ˵Ҫˡ
֪ˡ׶̧ͷҵĻΪʲô¶΢ЦͼҽѧáٵǉѧҼסˣõġ
šʲôõİ
صѧϰɣҪ˼ʲôأ׶˵




7¡ۻ


ʹ˸еȫûдⷨ
ô뵽أ
ʹ˸еչʵʵ飬
αֵأ
ҪôҲ뵽Щأ
[1]

71ͼ

711׶

ж꼶Ķ졣
𣿡
ڷѧͼڹõλ׼ʼʱ׶˹ҵǰһֳֽվš
ʲôʡ
Ӵľʦ⡣ش
ֽôдš

71

0101
1Ļ1(C<1>1)

012(0(12))
 ((01)2)
2Ļ2(C<2>2)

0123(0(1(23)))
 (0((12)3))
 ((01)(23))
 ((0(12))3)
 (((01)2)3)
3Ļ5(C<3>5)

0123n
nĻм(C<n>)

ĿóҪֱһͺˡұ߽Ŀ뿪ֽϱ߱Թ
มҪֱһ㣿ҪдҪصԼάʵĳȣҪʽûдܴ顭𣿡
˵
Ц͵ΪֹҵĵƹʽͿˡ
һ£׶ȣʲôʱõģ
ݵʦ칫ҵʱȻе͵ܡѾȷʵˣͷʼɣҵĵط룬ݰݡ
׶Ȼӻ֣ŵƶߵλҵĿû뿪һÿѾҶͩԶĵطһƬ죬Ȼʣȴ൱غ䡣
Һ׶ͬǸж꼶ѧǵѧʦľʦʱǣȻʦе֣ƺǵġ
׶ȵѧܺãȻҲȴӮÿͼʽȤʱͻչҵԶǦҵıʼǱдȻʼݽʱҲ˾ò졭
ϲ׶ר˵Ҳϲ۳˼۾䣬ܴ
ЩȻصϰɡڴ˼𰸣ƺѾƹʽˣĻܿͽˡ
Ҫһ°ɡ
010120123ʽӣţΪд1Ļ1֣2Ļ2֣3Ļ5֣ҪŷĿ0123nʽŷ
nʲôأʽ0123n0ʼʾn1nԴ0123nʽӺ()ĸ
ŵĹʲôڼӺҵʽӳΪ߸ȡһҲǻ(01)(0(12))2ĺ(Ƶ)ǲ(012)3ĺ͡
ǰۣȴʵʼĿдn123״Ծȴn4ʼϵضࡣ

01234(0(1(2(34))))
 (0(1((23)4)))
 (0((12)(34)))
 (0((1(23))4))
 (0(((12)3)4))
 ((01)(2(34)))
 ((01)((23)4))
 ((0(12))(34))
 (((01)2)(34))
 ((0(1(23)))4)
 ((0((12)3))4)
 (((01)(23))4)
 (((0(12))3)4)
 ((((01)2)3)4)

Ȼ14֣Ҳ4Ļ14֡

дʱֵ򣬿ҵйءŷĵƹʽˡ
ٳʵ󣬽ǹ廯еӺnʱ򽫡ŷΪC<n>ղżӺ4C<4>14ĿǰΪֹ֪C<1>1C<2>2C<3>5C<4>14ҲC<0>1ȥгͻı

n   01234 
C<n>112514

C<5>ӦҲøôһǡC<n>ĵƹʽˣĿǡnC<n>ıչʽ
ֿʼƹʽʱһλŮͼҵſ˹
ǵٵ١


712ٵ

ѧٵܵҵԣŻŵؿڣѾʼùˣ̫𣿡
ٵǸһ꼶ѧ֮ҵѧãСССèһңһЩѧ⣬ֻԲ⣬ҲϵʣȻеˣҲš
ܼ𣿡
᲻ᣬûϵֻеʶѡٵٱҡֱߺŵͲˣԵҪȥʱ١Ҳ𣿡
ǰӦûȹԱУΪֹҪһȥ𣿡
͵͵ؿ򴰱ߣ׶ţעֽϣңԿı飬Ҳûκζ
õģͬȸˡ
ٵɵ̤ȽŲھ֮תֱ߳ͼңڳȥһ˲͵͵׶һۡ


713ƹʽ

ôصŷĵƹʽ
045м4Ӻšϸ룬ҪǡŷЩֱû塣Ҳ˵

((01)(2(34)))

ʽȡ

((AA)(A(AA)))

ΪҪƹʽͱ뿴ŷĹ죬ȻҳԡʽĸӺţȻ3Ӻµ״Ҳ˵

 ((AA)(A(AA)))------Ӻĸ

ģʽҲ

 ((AA)(A(AA)))
  1Ӻ   2Ӻ
ţˣļӺšҲҪעŻӵļӺʽΪڶļӺţʽӻļӺŷֳʽӣӺŵλô߿ʼ˳ƶĻʵ4Ӻŵʽӿֳ4֣ļӺ<>һ

((A)<+>(AAAA))
((AA)<+>(AAA))
((AAA)<+>(AA))
((AAAA)<+>(A))

(AAAA)һûšȻ3ϵĺͣӺŵĸԽԽ٣ԿԴ֮ǰʽţƺƹʽˡ
ĸӺŵʽҲ˵

(AAAAAʽ)



(Aʽ)<ֱӦ>(AAAAʽ)
(AAʽ)<ֱӦ>(AAAʽ)
(AAAʽ)<ֱӦ>(AAʽ)
(AAAAʽ)<ֱӦ>(Aʽ)

￪ʼչɡĸnӺŵʽӵŷC<n>ʾĻC<n>ĵƹʽ
ֱӦڡ˼ǶӦĻn4״ҲC1ʽʱC44ĺ͡

C<0>C<3>C<1>C<2>C<2>C<1>C<3>C<0>

ҲC<4>дʽӡ

C4C<0>C<3>C<1>C<2>C<2>C<1>C<3>C<0>

ܹ廯ˡ

C<n1>C<0>C<n0>C<1>C<n1>C<k>C<nk>C<n0>C<0>

Ưʽˣʹæýṹܸ֡

C<0>1
C<n1><k=0n,C<k>C<nk>>(n0)

ܺãƹʽˡ
Ͽɡ

C<0>1
C<1><k=00,C<k>C<0k>>C<0>C<0>1
C<2><k=01,C<k>C<1k>>C<0>C<1>C<1>C<0>2
C<3><k=02,C<k>C<2k>>C<0>C<2>C<1>C<1>C<2>C<0>5
C<4><k=03,C<k>C<3k>>C<0>C<3>C<1>C<2>C<2>C<1>C<3>C<0>14

112514һʼʵǺϡ
ղ׶˵ġƹʽͿˡĽ׶Σ滨˲ʱ䰡

ǱУʱ䡣

ԱʦѴҡʦһŽȹױ̫۾ɫ۾ƽĹԱңʱ䵽˾ͻϢسͼ룬ѴұУʱ䵽ˡȹԱʱһ
ร˵׶أ
תͷһ£Ѿˡ

C<0>  1
 C<n1> C<k>C<nk>(n0)


72ڻؼҵ·Ͻ廯

ѧ廯ʲôأٵ˫ۡʵҷʡ
ٵٲվȥȻ֮ҹ׶ȣҵҲҲʧӦǻȥˣ֣Ѿ⿪˴ľʦ⣬ҪȥҲҪȴкɡ
ɫȻе谵·δסլĸС·ϡǴѧУվ̾룬ٵƽȻıģڻؼҵʱȴ᲻˼ߣҲֻĲ
Ҫһķ˵廯ףѧʽˣ23־ֹɵĹʽĹʽ任ɶnԳĴԹʽͳΪ廯
nԳĹʽ𣿡
ԣǶ23ֵָĹʽ޸޷234ȵһһо٣Ӧ˵Ȼо٣޷ֳȫúбnķʽȻnԴκԳǡԳĹʽáԳʾҲԡ
n
ڹ廯ʱ򳣳µıҲ˵ǡγɵĹ廯

ٵͻȻһ硣
𣿻˵ûΧΧ
ǵģϴҴææشӼౣıַ졣
㣬ĻðɡҰԼΧø
лллۣůͱѧ˰ɡ
ûϵûϵ
ԲҪܡƽ֡Χͺˡ
̫ˡ
ףѽǡҲǸ˼Ÿҡֻ֣Цš ˵˵ղŵġԳĹʽٽøһ𣿡ٵٸϽ߻ֱվá
õĺõģΪ·ʱû취дʽ˵ʱĻǵBeansͰɡ
ʱ䣬ʱ䡣ٵͻȻӿŲ׷ңΧŲΧǳɰ
ѧ㡫תͷҵĵٵ³ɫϢ


73ڡBeansʽ

ڳվǰġBeansһߺſȣһչʽ
ʽ

(xy)<ƽ>x<ƽ>2xyy<ƽ>

õģǶxyĺʽɡ
ţǽxyʽƽչӱֳ¸ʽʽ

(xy)<>x<>3x<ƽ>y3xy<ƽ>y<>

ﶼû⣬Žʽָ廯ҲǲƽηǡnηĹʽҪ(xy) nչʽ˼

72
nΪ1ϵչʽӡ

(xy)<nη>

ڹ廯֮ǰȽ֪ľ֪ʶһ£ٳʵȻ۲ҲȷԼǷѾˣԽʯ(xy)<nη>n1234룬ͻʽӡ

(xy)<1η>xy
(xy)<ƽ>x<ƽ>2xyy<ƽ>
(xy)<>x<>3x<ƽ>y3xy<ƽ>y<>
(xy)<4η>x<4η>4x<>y6x<ƽ>y<ƽ>4xy<>y<4η>

Ȼ廯Ḷ́ڿʼҪľʽӡ

(xy)<nη>x<nη>y<nη>

Ѿ֪x<nη>y<nη>ֻ֮Ҫx<nη>y<nη>ġͺá
ԲҼǲסٵ˵
ԣҪҪ˼˼
˼ʽӰɡ

(xy)<1η>(xy)
(xy)<ƽ>(xy)(xy)
(xy)<>(xy)(xy)(xy)
(xy)<4η>(xy)(xy)(xy)(xy)
.
    .
    .
(xy)<nη>(xy)(xy)(xy)(xy)
        n

ҾͶˣǰ(xy)nΡ
ǰԵn(xy)˵ʱ򣬾Ǵÿһ(xy)ѡxyˣƩ˵ηǴ(xy)иѡ1xy˼ȫѡʽѡĲ<>Ǻš

 (<x>y)(<x>y)(<x>y)xxxx<>
 (<x>y)(<x>y)(x<y>)xxyx<ƽ>y
 (<x>y)(x<y>(<x>y)xyxx<ƽ>y
 (<x>y)(x<y>)(x<y>)xyyxy<ƽ>
 (x<y>)(<x>y)(<x>y)yxxx<ƽ>y
 (x<y>)(<x>y)(x<y>)yxyxy<ƽ>
 (x<y>)(x<y>)(<x>y)yyxxy<ƽ>
 (x<y>)(x<y>)(x<y>)yyyy<>

ȫгˣȻЩȫ

xxxxxyxyxxyyyxxyxyyyxyyyx<>x<ƽ>yx<ƽ>yxy<ƽ>x<ƽ>yxy<ƽ>xy<ƽ>y<>

ͱ

x<>3x<ƽ>y3xy<ƽ>y<>

Ҫʽӣ(xy)(xy)(xy)չġ͵Ļx<>3x<ƽ>y3xy<ƽ>y<>֡ĺ͡˵ĺ͡ɡ͵Ļʽֽ⡣
ԭˣڶˡܾxxxxxyxyxyyyЩзʽйԡ
ţรٵ١
ٺ١ߵͷ
ǼɣҪ(xy)ѡxy֮һôȫѡxѡмأ
ţһҪѡxĻֻ1
ûôxn1y1ѡأ
ţұѡyѡxұڶѡyĻn
ˣȷ𰸣ǽǹ廯xnkykѡм
ţn(xy)ĸĻkʲô
Ǻܺõ⣬kΪҪ廯ıʾѡyĸkΪ0knղ۵k0(ȫѡxѡ)k1(y1ѡ)Ρ
ǴnѡkΣΪѡ˳Ѿˣϡɡ

ԣϣyѡkxѡnkϵĻͻʽ

 ()<n,k>((n0)(n1)(n(k1)))/((k0)(k1)(k(k1)))

x<nkη>y<kη>ϵ
ѧ⡣ٵپ֣()<n,k>ʲôأϵĻ<,n,k>ɣĻһ
ǵģ()<n,k>nCkȫһѧϺܳд()<n,k>⣬дҲ()<n,k>ûйϵ
ã֪ˣһ⣬Ҽǵǡ

<,n,k>n!/(k!(nk)!)

ѧʽӲ̫һ
罫(nk)!ĲԼ֮ͻᷢʵһġƩ˵5ѡ3

<,5,3>5!/(3!(53)!)
      5!/(3!2!)
    (54321)/(32121)
    (543)/(121)
    ()<5,3>

һġ
õݽ׳˱ֻνĵݽ׳дx<nεݽ׳>Ǵӵn׵Ľݲ½ĻรҲ˵
<nεݽ׳>(x0)(x1)(x2)(x(n1))-------nʽ
           
ͨ׳nĵݽ׳дɡ

n!n<nεݽ׳>

ʹõݽ׳ˣͿԽ()<n,k>ֵøƯ
()<n,k>n<kεݽ׳>/k<kεݽ׳>

nѡkϵĸ

<,n,k>()<n,k>
        n!/(k!(nk)!)
        ((n0)(n1)(n(k1)))/((k0)(k1)(k(k1)))
        n<kεݽ׳>/k<kεݽ׳>


Ǹ΢ˣصɣѾõ(xy)nչʽˣΪ˽Աֳд΢߳һ㡣
(xy)n(ѡ0y)
(ѡ1y)

(ѡky)

(ѡny)
()<n,0>x<n0η>y<0η>
()<n,1>x<n1η>y<1η>

()<n,k>x<nkη>y<kη>

()<n,n>x<nnη>y<nη>

עÿһ仯Ĳ֣æ־ͻõеʽӣǶʽ

72
(xy)<nη>չ(ʽ)

(xy)<nη><k=0n,()<n,k>x<nkη>y<kη>>

һʼ֪չǲ׼䣬ԼֵʽľͲ̫ѼˣϰԼʽĻͻڲ֪мס֮ͲҪˡȻǷ˵ҲȤġ
ѧ˦ƺͻȻúˡ
粻ĻҲԽʾдk0ʱk1ʱk2ʱϰ֮ǰҪ
û뵽õϣʵʱϰѡͰʱһѳ˷ӡ̣ϰԼһû뵽ʽչлַʽõϡ
ˣ˼ӣ廯ǰһҪ㣬ﲻ͵n1234ȷϡ

(xy)<1η><k=01,()<1,k>x<nkη>y<kη>>
 ()<1,0>x<1η>y<0η>()<1,1>x<0η>y<1η>
 xy

(xy)<ƽ><k=02,()<2,k>x<nkη>y<kη>>
 ()<2,0>x<ƽ>y<0η>()<2,1>x<1η>y<1η>()<2,2>x<0η>y<ƽ>
 x<ƽ>2xyy<ƽ>

(xy)<><k=03,()<3,k>x<nkη>y<kη>>
 ()<3,0>x<>y<0η>()<3,1>x<ƽ>y<1η>()<3,2>x<1η>y<ƽ>()<3,3>x<0η>y<>
 x<>3x<ƽ>y3xy<ƽ>y<>

(xy)<4η><k=04,()<4,k>x<nkη>y<kη>>
 ()<4,0>x<4η>y<0η>()<4,1>x<>y<1η>()<4,2>x<ƽ>y<ƽ>()<4,3>x<1η>y<>()<4,4>x<0η>y<4η>
 x<4η>4x<>y6x<ƽ>y<ƽ>4xy<>y<4η>

ٵٽʽһȷ֮ͷ˵Ȼʽһֻ˾á÷һ뵽ǹ廯Ľ;ÿԽܣһҲû취¡
ţΪȡ޸Ĺʽnǹ廯ĹʽڸĲҲk廯
ǵģnkkһҪֱҲ鷳
Ҫnkkֿ˼Ҫ롺;nȻд0nȡƽ⣬һʼxָnʱyָ0СȻxָÿ1yָͼ1xָС0yָnҪ˼kмƽλá

k0xxxxxx
k1xxxxxy
k2xxxxyy
k3xxxyyy
k4xxyyyy
k5xyyyyy
k6yyyyyy

xyƶ
ûȫnη䵽xyϣƽ֡Χһ
ѧѧ㻹ǵⰡ


74ԼнɺĻ

ҹˣҲ˯ˣҶڷ侲˼C<n>ĵƹʽѾˡ

C<0>  1
 C<n1> C<k>C<nk>(n0)

ҽ볢һǾɺĽⷨ
׶ȺѰҹ쳲еһʱɺ˶Ӧȡ֮롺ɺ֮лơ
Ҵ򿪱ʼǱһѰһ߿ʼд¡
õa<0>a<1>a<2>a<n>֮󣬾ͽиϵa<0>a<1>xa<2>x<ƽ>a<n>x<nη>ʽݼ֣ɺȻĶӦϵɺΪһĶ

        С                            ɺ
a<0>a<1>a<2>a<n>    a<0>a<1>xa<2>x<ƽ>a<n>x<nη>

˶ӦĻͿԽһɺ֣ǽɺԱչʽ֣ͻõһıչʽ̾Ľ
Һ׶ʹɺ쳲еһԭϿҪɢУΪɺһߴһľ顣
ֽⷨ⿪ε⡣

(Cnչʽеͼ)
C<n> ɺC(x)
 
C<n>ıչʽ ɺC(x)ıչʽ

nӺŹɵʽC<n>C<0>C<1>C<2>C<n>
ٽ֮ɺΪC(x)xΪ˲лҵʽϱx<nη>ָnC<n>nӦC(x)ʾ

C(x)C<0>C<1>xC<2>x<ƽ>C<n>x<nη>

ɺĶ壬ΪֹҪԽûҪɺĹǺܼ򵥵ġ
ҪԵĲִڲſʼ
ӵеֻC<n>ĵƹʽѣһҪõƹʽC(x)ıչʽC(x)ġxıչʽʽӦòn
εĵƹʽ쳲ʱһʱȷʵɺгxȻ󲻶ϵءƶϵŽn
εĵƹʽC<n1><k=0n,C<k>C<nk>>൱鷳C<k>C<nk>ټ˦γɷġĺ͡ʽ

ţ
ĺ͡
C<k>C<nk>֮֡Ϊnʽ
ԭˡ
ԼԵٵ˵Ļˡ

Ҫnkkֿ˼Ҫ롺;nȻд0nȡƽ⡭

εĵƹʽ ҲƣC<k>C<nk>ı֮ΪnȻΪ˺͵ƽ⣬k0n֮䶯
֪ĵƹʽC<n1><k=0n,C<k>C<nk>>ֵģܺúæ<k=0n,C<k>C<nk>>ɡĺ͡ʽͿȽϵû
ϸ룬Щϻ֡ĺ͡

(xy)(xy)(xy)֡͵Ļx<>3x<ƽ>y3xy<ƽ>y<>֡ĺ͡չ

͵Ļչͻɡĺ͡
á
ؼƺǻˣԿɺĻɣܷʲô
ֻɺC(x)Կƽʲôأɺ¡

C(x)C<0>C<1>xC<2>x<ƽ>C<n>x<nη>

ƽĻ

C(x)<ƽ>(C<0>C<0>)(C<0>C<1>C<1>C<0>)x(C<0>C<2>C<1>C<1>C<2>C<0>)x<ƽ>

C<0>C<0>xϵC<0>C<1>C<1>C<0>x<ƽ>ϵC<0>C<2>C<1>C<1>C<2>C<0>
ù廯˵ٵ˫۾C(x)<ƽ>x<nη>ϵ
Ŀռֻʣдֵɳɳ
ˣx<nη>ϵ

C<0>C<n>C<1>C<n1>C<k>C<nk>C<n1>C<1>C<n>C<0>

ҪעǵĲ֣C<k>C<nk>УߵkұߵnkСk0nķΧƶ
Ϊֹд൱߳׶ʹæ˵xϵ

 <k=0n,C<k>C<nk>>

C(x)<ƽ>ʽӵġx<nη>ϵC(x)<ƽ>ʽӾͻɶغ͵ʽдɡ

C(x)<ƽ><n=0,<k=0n,C<k>C<nk>>x<nη>>

ˡ
ˣ
Ưġĺ͡<k=0n,C<k>C<nk>>ˣ֮Ӧÿõƹʽ򻯣ɵƹʽá

 <k=0n,C<k>C<nk>>

û

C<n1>

Ҳ˵
ԽɺC(x)ƽˣ C<k>C<nk>C<n1>滻ɡ

C(x)<ƽ>C(x)<ƽ><n=0,<k=0n,C<k>C<nk>>x<nη>>
C(x)<ƽ><n=0,C<n1>x<nη>>

มغͱһĺˡ
һ£C<n1>ıǺx<nη>ָ1
šˣ״쳲еʱҲйֻҪĲֳxͺã߳x

xC(x)<ƽ>x<n=0,C<n1>x<nη>>

ұߵxС

xC(x)<ƽ><n=0,C<n1>x<n1η>>

n0ĲΪn11Ϊϱָ

xC(x)<ƽ><n11,C<n1>x<n1η>>

Ȼn1ȫûn

xC(x)<ƽ><n1,C<n>x<nη>>

ܺãұߵ ͼɺC(x)ˣֻҪC<n>Ĳּ

xC(x)<ƽ><n0,C<n>x<nη>>C<0>

Ͱnˣ

xC(x)<ƽ>C(x)C<0>

C<0>1룬ʽ

xC(x)<ƽ>C(x)10

óC(x)Ķηʽx0Ȼõʽ

C(x)(1<14x>)/2x

š

˳
ɺĻƯġĺ͡Ȼ󵼳չʽû뵽ɺĻôá
һΪʲôһһɺ<14x>Ĳô£ƺкա
ô˵nѾˡ
ҵɺC(x)ıչʽ
֮ǽչʽݼչͺá


75ͼ


751׶ȵĽ

ѧͼ׶ҵԡ
ԭõƹʽġ׶ȿڣ;ı䷽ˡ
ףõƹʽ𣿡
ûõƹʽ⣬Ϊҵ˸õĶӦ
(õĶӦ)
Ҵ򿪱ʼǱ׶Ѹٵд
n4

((01)(2(34)))

ϸ۲Ļʹŵĺ롻ҲԸԭ

((01(2(34

ſԸԭģǡӺơ
ԭˣֻҪڼֵʱŵĺͿˡһ»ش룬Ȼˣû뵽((AA)(A(AA)))Ը򻯡
׶ȵ촽΢΢¶΢Ц
˵øһ㣬ָͲҪֱӱɡ



ǿԸԭģֻҪڼӺŵ֣4ҪдҲࡣ
ԭˡ˵
˵ŷǡǰ롻롺Ӻšϣn4˵4ǰ4ӺŵУ8š



Ȼ轫4ǰ롣



Ȼٽʣ*ԶɼӺš



8ӺŸ4ѡΪǰ4Ͼ()<8,4>n4廯Ǵ2nУӺŸn)ѡΪǰnϣҲ()<2n,n>ϵĻҲͬͼз··ߣµSʼϵGͷָĵ·ӦС

 [ͼһ44ıÿΪΡ½һΪSϽһΪGһеÿдһһеÿϷдһ֮½رͷңSһֱ赽G]

ô
һ¡Ҵϲ׶ȡ
׶ȣе֡ΪⲢ8֮ȡ4Ʃ˵㽫ӺŸȡ4Ҳų



Ӧ㻭ͼϾ֪ˣͼܾСĵطٵյ㡣

 [ͼһ44ıÿΪΡ½һΪSϽһΪGһеÿдһһеÿϷдһ֮½رͷңSһֱ赽G㡣֮SұһĽ㴦򣬲Ϸ45Ƚǵнϻ]

ϻ׶챧Թһû˵갡

򡡡򡡡

һû˵갡ӺʱżӺܳơ
Ӻʱ򣬾ͻ˵һҲͼͨ״ͨSGķŻC<n>
ƵĻSGķ()<2n,n>
ôSG֮ͨһϵķжأ
һĵطΪPͨP֮ǰķ䣬бߵɾӣPG֮ԭǡĻ͸ĳɡԭǡĻ͸ĳɡҲ˵յ㲻GG
GGеͶӰ㣬˵ǽɣ
˼ĻͨķͻʹSGķһһԶӦ2nĵ·ɺn1ĵ·ϣҲǱ()<2n,n1> 

    [ͼһ44ıÿΪΡ½һΪSϽһΪGһеÿдһһеÿϷдһ֮SұһĽ㴦򣬲Ϸ45Ƚбߣб߾ĵнϻ֮ڱҲಹһиӣȥһ²ĸϽǵĵΪG㡣֮½رͷңSһֱ赽G㡣]

仰˵ʽ

C<n>(SGķ)(SGķ)

Ǽˣ㣬ʹõݽ׳˰ɡ

C<n>()<2n,n>()<2n,n1>
2n<nεݽ׳>/n<nεݽ׳>2n<n1εݽ׳>/n1<n1εݽ׳>ʹ()<n,k>=n<kεݽ׳>/k<kεݽ׳>
((n1)2n<nεݽ׳>)/((n1)n<nεݽ׳>)(2n<n1εݽ׳>n)/(n1)n<nεݽ׳>ͨ

ߵ֣ͨǵڶеѶȻֻҪ׵ݽ׳˵ĺͻǲһ¡
εģǽ(n)β͡

(2n)<n1εݽ׳>(2n)(2n1)(2n2)(n1)(n)
 (2n)<nεݽ׳>(n)

Ȼĸεģǽ(n1)ͷ

(n1)<n1εݽ׳>(n1)(n)(n1)21
(n1)(n)<nεݽ׳>

C<n>ɣֺͨ󡭡

C<n>(((n1)2n<nεݽ׳>)(2n<n1εݽ׳>n))/((n1)n<nεݽ׳>)
(((n1)n)(2n)<nεݽ׳>)/((n1)n<nεݽ׳>)(2n)<nεݽ׳>
(1/(n1))(2n<nεݽ׳>)/n<nεݽ׳>) 
(1/(n1))()<2n,n>n<kεݽ׳>/k<kεݽ׳>()<n,k>

õnӺŵʽӵŷʽ¡

C<n>(1/(n1))()<2n,n>

ã͸һˣ㿴ɡ

򡡡򡡡

һΪ׶ȵļ򵥽ⷨе𾪣һ߼㡣

C<1>(1/(11))()<2,1>(1/2)(2/1)1
C<2>(1/(21))()<4,2>(1/3)((43)/(21))2
C<3>(1/(31))()<6,3>(1/4)((654)/(321))5
C<4>(1/(41))()<8,4>(1/5)((8765)/(4321))14

ȷʵ12514
׶ҵĻ¶΢Ц

71

C<n>(1/(n1))()<2n,n>

λˡ


752 ɺ

ȻǱ׶ӲҵŵĽⷨǺ𾪣ʹɺ𣬿ֻıչʽҲûҵȷ𰸣ǲսأɺĻĸжѾɢˡ
е㲻ġ
׶Ȱڳеŵıߴңûϵ˵˵ɣƹʽȻأ
˵볢ɺĽⷨɺĻƯĻĺ͡ٵηʽ󵽴ɺıչʽȻִɺĹȣȴزеĹȡ
ǳزġ
ʲôʽӣ׶ʡ
û˵
ţʲôʽӣҵ
û취ֻڱʼǱдʽӡ

C(x)(1<14x>)/2x

ţ⣬Ĳ<14x>Ĳ֡
Ҳ֪ǿﰡ
׶Ȳҷ˵ȥ
ȴӡĲ˼
׶ȿһʽ֮۾ƺоʲôϷʳָָȻתȦ㡢㣬Ȼ۾
صɣɺC(x)ʽӰɡ

C(x)C<0>C<1>xC<2>x<ƽ>C<n>x<nη>

Ҳ˵x0ʱ򣬺xȫʧC(0) 0ʱٻص㷢ֵıչʽɡ

C(x)(1<14x>)/2x

C(0)ôأ
УΪ0ǳC(0)޴󡣡һشѾ侲ˣΪ׶ôƢô
ԡ׶ȻҡͷȻһ޴󣬵һǲ̶ġC(x)ġΪC<>(x)ΪC<>(x)

C<>(x)(1<14x>)/2x
C<>(x)(1<14x>)/2x

ͻΪ˲ɳͽĸƹȥ

2xC<>(x)1<14x>  
2xC<>(x)1<14x>  

x0ʱ߶01<14x>21<14x>Ż0ôأ
ٿ֪C<>(x)ǲϵġ
ŰɣȻûȥѧɺû취˵ûбҪȥC<>(x)ˣʽֻҪע⼯C<>(x)ͺãΪأ
Ǵ<14x>ɡ˵
Ѿظң׶΢΢һЦ

ɺC(x)ıչʽ

C(x)(1<14x>)/2x  


753Χ

ʱע⵽ٵվͼҵڣһҺ׶ȣֽǰǴʲôʱʼվŵأ
ٵ֣ƽͬǱıضϻ¶顣
ѧллˡ
ٵƽ˵Ų˸ȻֽңкõΧ
ţûðɡ
ţû£ΪѧΧֺһ
ٵٱ˵߽ת׶ȣҲſȥ׶ԶǦʵ̧ͣͷֽƳһۺٵ٣ŮԵض
ûκ˵

롣

ٵ١һҡ
͸ˣ֮Ҳѧٵи񣬻߳ͼңڵʱֻعͷٴ
ʱ׶Ѿֽţ׼㡣
뵽ʲô𣿡ʣȻǹ ¡
׶û̧ͷһдʽһ߻ش

š

ף

š׶Ծûֹͣ㡣

ҿ˿ӲֽȥңΧƺʲôóŷ൱װɫƬΪʲô׶Ȼע⵽пƬأ
ŵٵµļѶϢ

ллůΧٵǉ
P.S.ҪԼȥBeansร


754Ĺؿ

ǻصϡ
ɺC(x)ıչʽʾ

ɺC(x)ıչʽ

C(x)(1<14x>)/2x  

Ҫô<14x>ˡ
ƺҲһҪôˣ׶ȣõC(x)ıչʽ֮󡭡쳲һʱôģ
C(x)ıչʽֻҵx<nη>ϵ򵥵˵չݼ׶˻ش
<14x>鷳˵Ҫô<14x>أұԹš
Ҳֻչݼˣ轫ϵK<n>Ϳչ׶дʽӡ

 K<0>K<1>xK<2>x<>K<n>x<nη>
 <k=0,K<k>x<kη>>

ȻɺC(x)ʽӡ

C(x)(1<14x>)/2x

Խĸʽӡ

2xC(x)1<14x>  

C(x)<k=0,C<k>x<kη>><14x><k=0,K<k>x<kη>>ͻɡ

2x<k=0,C<k>x<kη>>1<k=0,K<k>x<kη>>

2xƵ棬ұߵk0档

<k=0,2C<k>x<k1η>>1K<0><k=0,K<k>x<kη>>

ߵɴk1ʼ

<k=1,2C<k1>x<kη>>1K<0><k=0,K<k>x<kη>> 

߼С

<k=1,2C<k1>x<kη>><k=0,K<k>x<kη>>1K<0>

áˣҪı͵˳˵ΪҵʽӾʡԡ

<k=1,(2C<k1>K<k>)x<kη>>1K<0>

ʽǶxĺʽԽߵϵȽ֮󣬾ͿԵõKnCnĹϵʽ

  01K<0>       Ƚx<0η>ϵ
2C<0>K<1>0 Ƚx<1η>ϵ
2C<1>K<2>0 Ƚx2ϵ
.
    .
    .
2C<n>K<n1>0Ƚxnϵ
.
    .
    .

֮õ

K<0>1
C<n>-K<n1>/2(n0)

ҲK<n>ĻҲԶõC<n>Ĺؿ<14x>չˡ


755

׶ƺȲ˵
ôĹؿɣK(x)<14x>ȻĿ󡭡

K(x)<k=0,K<k>x<kη>>

(K<0>K<1>K<n>)￪ʼأ
׵ĵطʼɡ˵
ร֪Ҫô𣿡
Կx0ɡϻش𣺡Ļ<k=0,K<k>x<kη>>˳ⶼҲǻ

K(0)K<0>

ûȻأ׶ʡ
xҪô𣿡ҷʡ
ǣҪϿýĻ׶е㲻õػش
ʲô
΢֡K(x)x΢ֵĻоͻ任K1

K(x)K0K1x<1η>K2x2           K3x3          Knxn
	       	          			
K(x)1K<1>2K<2>x<1η>3K<3>x<ƽ>nK<n>x<n1η>

ԡ

K(0)1K<1>

֪ΪʲôҪд1˰ɣΪ΢ָֻ½ΪĹɣˣK(x)΢ֻõʽӡ

K(x)21K<2>32K<3>x<1η>n(n1)K<n>x<n2η>

Եx0ʱʽӡ

K(0)21K<2>

֮ͲϵظK(x)΢nK<(n),>(x)ʾĻ

K<(n),>(x)n(n1)(n2)21K<n>(n1)n(n1)(n2)鷳

Ϊ̫ˣõݽ׳д

K<(n),>(x)n<nεݽ׳>K<n>
(n1)<nεݽ׳>K<n>1x<1η>

(nk)<nεݽ׳>K<nk>x<kη>


x0ʽ

K<(n),>(0)n<nεݽ׳>K<n>

ҲK<(n),>(0)ԱʾK<n>򵥵˵̩չʽ

K<n>K<(n),>(0)/n<nεݽ׳>

һ䡣
׶ȴһ
ţ޷ȥˣѾû·ˡ˵
Ϊʲô˵أѾݼ׽סK(x)ˣͨĺʽ׽ɡ
׽
ʹýĻ΢֡
˵껰׶ȶգգۣһǴı֡
K(x)Ķ塭

K(x)<14x>

Ҳ˵ƽ1/2ηԡ

K(x)(14x)<1/2η>

һעɣһ߷΢֡

K(x)(14x)<1/2η>
 K(x)2(14x)<-1/2η>
 K(x)22(14x)<-3/2η>
K(x)243(14x)<-5/2η>
K(x)2654(14x)<-7/2η>
.
    .
    
K<(n),>(x)2(2n2)<n1εݽ׳>(14x)<-(2n1)/2η>
K<(n1),>(x)2(2n)<nεݽ׳>(14x)<-(2n1)/2η>

x0γʽӡ

K<(n1),>(0)2(2n)<nεݽ׳>

ٰѸղݼõʽӣ˵û취ȥǸʽón1˼

K<n1>K<(n1),>(0)/(n1)<n1εݽ׳>

ʽӣԵõʽ

K<n1>(2(2n)<nεݽ׳>)/(n1)<n1εݽ׳>

͵õK<n1>ˣȫ·㻹ǵK<n>C<n>Ĺϵ

Cn-K<n1>/2

ּ֮ˡ

Cn-K<n1>/2 
(2n)<nεݽ׳>/(n1)<n1εݽ׳>

ĸԴ(n1)<n1εݽ׳>(n1)n(n1)1(n1)n<nεݽ׳>Ρ

(2n)<nεݽ׳>/(n1)<n1εݽ׳> 
(1/(n1))((2n)<nεݽ׳>/(n)<nεݽ׳>)
(1/(n1))()<2n,n>

͵õC<n>

C<n>(1/(n1))()<2n,n>

ã͸һˣõͬʽӣҲǴɺĹȻˡ
׶㵽¶Цݶ˵
ӭ


756뾶ΪԲ

һˡӦҪ˵ллŶԡ˵
൱Ȥ˿ֵСʳָ

ҿ׶ȣǡȻе³ȴ侲ر飬ҹȻ׶ȡ

׶΢۾վ

Ϊ˼ֻɡ

Ҳվ

(ʲô˼)

׶ֱ֣Сͣ׶ȴ׵ָ䡣

(ů)

ǣǰĿյƶ
׶ԻͼķʽҵΧ߹
һ
һ
ĽŲ
׶š
ѧͼҳûˡ
ֻü΢ĽŲ
׶ұͬĵطԲϣⵥλԲɣ
ҵ˵ʲô
׶ȡšһͣ½Ų˵ֳȼ1ĻǵλԲشȻ۾

޷ġ롻Ҳϣġ

顣
׶ſۡ
ʹ뾶㡭˵һ룬׶Ⱦؽ
ʹ뾶㡭ǻֿ𣿡
˵ŵ׶Ƚֱҵ۾ľ
ʲôҲ˵
׶Ҳû˵ʲô
ʹ뾶㣬ԲԲѾɵԲ
Ȼҡ
ǡ
Եء
ؽտ

ǱУʱ䡣

ȹԱ
ǵľһӡ
ֱֳĺΪֹ

ҡıʼǱ

Һ׶һC<n>112514Ϊ(Catalan number)(֮Ҳƿ)˼ƯĻĺ͡Ϊۻ(Convolution)(֮Ҳ޻)
ɺӦĻܰѡۻС͡ԭɺõĺӦҲǽa<n>b<n>ۻa<n>*b<n>ʾĻγ¶Ӧ

            С ɺ
a<n>a<0>a<1>a<n> a(x)<k=0,a<k>x<kη>>  
b<n>b<0>b<1>b<n> b(x)<k=0,b<k>x<kη>>

ҹڷ˷ܵӦйȡġۻǡɺȡġ
ĶӦ



8¡


    ͹Ϊ֮һȺڶԻ
ΪһͻȻĬ
ΪֵԼٶ˵֮ǰ
˵Ļ
˶͹С(ǲһ)



81Ѱ


811ٵ

ѧ
ѧᣬվУſڣٵ˹
ԭﰡͼқ]㣬Ϊʲô飬ѧҪȥĻһ𡭡ףǣ
ҽŵĿƬƵſƬϡ

 (ҵĿƬ)

 H<><k=1,1/k> 

ٵǱСһ졢һ꼶ѧãƽСһҵߣǸʱͼһùĺû顣ܰ˵ȻأżҲ¶̾飬һͷ̷˫۾Ҿ൱ɰ
ǡʲôٵ̧ͷ
ţо⣬ǴʽӳȻѰҡȤĶϰ⡣
Ծһı顣
ʽӾǲűɭһܲڵҳſƬǴӴľʦġ
ҳǡٵٶȿ˿ҵĿƬ
ţſƬԼ⣬Ȼٽ⿪
ԭˡѧѾʽҵ𣿡
ûſƬ֪һ㣬ʽH<>ʽұߵĦ<k=1,1/k>ǡ

ٵͻȻһס͡
ǸѧʲôҪ˵Ҳһ𣬡Ѱ𣿡
ʲô˼
оҲ뿴ΪҵΪֹûԹҪһοŬھ
˵ĵٵٰڳڶӡ
ȻԣҵʲôȤĶдľʦɡ
ףõ̶ֳȰɡ
æҡͷٵٻһг
ſƬȸ㣬ͼۣ뿴ɡ
ãһ͵ġ
ٵ˫۷˫ȭա
ѧѧҡ
ٵٵͻȻֹͣˣעҵı󣬽Сءѽһ
һعͷ׶վ


812׶

õˡ׶΢Ц
УǰŮм䡣
׶Ǹͬࡢж꼶ѧҲǸ볤۾൱ŮѧܺãȻдҵıʼǱͻȻʼѧһЩܱ뷨顭
ٵͻȻѧڵ˰ҡҺˡǸȸˡһͷ˺벽
š
׶ؿٵ٣ٿңֿٵ٣Ȼ۾΢Цúͻ˵
ûϵٵǉһ˻ȥͺˡ
׶ĵٵٵͷŴҺ͵ٵ֮߹
ٵڱĵʱ΢΢ͷգգ۾ֻܿ׶ȥ
Զȥ׶ûлͷֻҪشĿȥĵٵһֻ˻ӣ󡭡ʧתǡ
һʱŲΪ׶߹ʱҵĽš
Ƿǳء
ʹ


82Իеͼ

죬ѧͼ˲ࡣ
Σʡ
ٵ¶Ҫ޳һı򿪱ʼǱֻдһʽ

<k=1,1/k>(1/1)(1/2)(1/3)

ѧҵѧȻС
ץʽӱ˼ˣʽû
ѧȻҺҵȤĶȫ֪֮Ҫʲôźá
޵ʽȻԸоʲôҪʵʳȴѡٵٵսֵͣڿʼǾһѰ
ף˼㱦ʱ䡣
ûϵǾǰɡ


821ֺ

ȿʽӦ<k=1,1/k>ʽѶĲǡޣ󣩵ĵطɡ
޴ǡ
ޣ󣩲ǡͨʵͲޡ
𣿡
ǵģд<k=1,1/k>ʱ򣬾Ҫɡk1۽ޣ1/kۼӡ˵ɡ޺ĳطһȻkǰȥ˵ǲȷģ<k=1,1/k>Բֺͦ<k=1,1/k>ļ޶ġ

<k=1,1/k>lim<n,<k=1n,1/k>>

Ǹlimǡ
limitǼޣѧϵĶֻܳ򵥵˵ٶa<0>a<1>a<2> a<n>Ǳʾnڷǳʱ򣬡a<n>ֵΡʽӡnǳʱa<n>ܻ᡺ֹ󡻣Ҳ᡺ʱʱСܡӽһֵʽڽӽһֵʱa<n>ᱻ壬֮ܶǱʾ a<n>ġĿص㡻ᵽĳĿصļƴ˼
šѣnڷǡʱ򣬻ôĲһö
ţ̫˰ˣǺһ֣ԲʽдȶԡĿصǱġиӡͺãȻǱģͲһֱܿ⡣ϾȥֵӲֺͿʼ˼n޵ļ޲ȷķ
˼Լֺ͵Ĳһǲ̫
Ҳ˵ֻǼ

 <k=1,1/k>(ʹkʽ)

ǲֺ͡

 lim<n,<k=1n,1/k>>(ʹnʽ)

Σ˽𣿡
ţ޺nǲһġΪnǱԺ͡޲һ𣿡
ԣȫһȷʵnǱʾ޵޲Բn룬Ϊn޵Ԧ<k=n,>ϲŻ޸ӣҲһܵõ<k=,>޸ӣͲһõǸո΢ġֹر󡻻ʱʱС״Ŀصͻ᲻̶̶ֵͲ⣬Ŀص㲻̶޷ɢ޸ʱһҪǳעĵط
õġѾ֪ҪעˣҲ֪˷ɢԸйصĻʽҲһǹ̶ֵ
ǱҪעĵطʽӶõɾںš(1)(2)أ

 (1/1)(1/2)(1/3)(1/n) (1)
 (1/1)(1/2)(1/3)(2)

ʾġӦ(2)ɡ
û(1)(1/1)(1/2)(1/3)(1/n)ġǱʾֻΪдԲд޵ģֵǹ̶Ĳֲ(2)(1/1)(1/2)(1/3)ġֵľlimͬ˵ֵ᲻̶޵ġġ˼ȫһҪرע⡣
һġŲһ˼ء


822Ȼĵطʼ

ร̸޵Ļˣڿʼǰϰ޸ĺͲСΪҪϰߦȽnΪ12345ʽӾг

<k=11,1/k>1/1 
<k=12,1/k>(1/1)(1/2)
<k=13,1/k>(1/1)(1/2)(1/3)
<k=14,1/k>(1/1)(1/2)(1/3)(1/4)
<k=15,1/k>(1/1)(1/2)(1/3)(1/4)(1/5)

ڿʼ㲿ֺͰɣҪע⦲<k=1n,1/k>ֵǡnġԼʹдH<n>ҲǿԵģH<n>Ķʽ

H<n><k=1n,1/k> (H<n>Ķʽ)

˼Եһ£nҲ̫
ţὫԼĵطǵٵٵŵ㣬51000ֻҪnֵǾ̶ģ<k=1n,1/k>ʽӵֵǹ̶ģǡn˼ԿдnΪǵH<n>ĻH<5>ͺH<1000>һֻϵ⡣
ΪʲôHأ
ΪƬдH<>Բֺ;H<n>
ԭˣ˵дΪH<n>ȻnˣkΪʲôȴʧˣ
Ϊ<k=1n,1/k>еkֻڦʹõıĵطkֱΪԼ˼ڦбԼıҲһʹkɣԼϲ֣ijklmnЩʹáiǱʾ<>λԻɻҵʱͲҪãƽnԼﲻУΪnѾ˼ˣѦ<k=1n,1/k>дɦ<k=1n,1/n>Ļ˼ͻú֡
õģ֪ˡ˼ѧ˵
ûϵ֪ĵطȽ׽ȥ
ӶЦ


823

оH<n><k=1n,1/k>йصĲ֣ΪԽʯȷ𣿡

n1ĻH<n>1

ȷġΪH<1>1ǵȻġԭˣȻĵطʼ԰ɣ
ûסˣ𣿡

еnH<n>0


жǷѧųΪ⣬ùӢʽдǣ𣿡

еnnʱH<n>Ȼ

ǵģûnʹиӡ
ǵģӾͻ󣬡nʱH<n>Ȼ⣬ҲʽдʽӣַʽȽܡ

еnH<n>H<n1>

ȷʵ𡺵nʱH<n>Ȼ󡻣H<n>H<n1>Ƚܰš
Ҿصȴٵٵ˼
֪ˡʹòȺŵġڡıǲͬİɣӢһ㶯ʺbeһ
ס
ٵٵĻһЩ󡻺͡󡻵Ĳһ㶯ʺbeʣԭˣɣ֮ǰľʦ΢׷Ѱб仯̬Ĺ۵벶׽иϵʽĹ۵㡭ǡ̵ĶĶ塻Ļ⡭
ѧô𣿡
˵ԺҲ뵽ҲֻֿҪһʽȽܡѣ˵ٵ㵽ǡ
ʲôٵͷ۾µµתš
û¡ǰɣһ𣿡

еnH<n1>H<n>1/n()

ΪH<n>ԷĺģüַҲǵȻġ
ܿϧˣH<n1>H<n>1/nұߵķĸǴģҪʽӣĸnn1Ż

еnH<n1>H<n>1/(n1)

סԭˣѧ̫ˡٵٿʼұԹ
ǸǸҪúõȷϹС
ô˵û졣
ôH<n1>H<n>׻ôأʹH<n>Ķʽ㶯Կ
õģš

H<n1>H<n><k=1n1,1/k><k=1n,1/k>

HnĶʽѦд

(1/1)(1/2)(1/n)(1/(n1))(1/1)(1/2)(1/n)

ãˡȻ󡭡š˳ı䡣

((1/1)(1/1))((1/2)(1/2))((1/n)(1/n))(1/(n1))

Ϳ˰ɣѧ

1/(n1)

ãúܺãɵٵ⿴
šΪH<n1>H<n>𣿡

еnnʱH<n1>H<n>С

ûûܺãʽĻҪôдأ
𣿡

еnH<n1>H<n>H<n2>H<n1>

ǳã
ȥ1/21/31/4֡СĸоáСʽ֡


824ȫġ

ٵ٣һжʽʾǺҪģһЩȻҲҪддһϰʹѧԵķ
ãѧ֮ǰ˵ѧ󡭡ٵһ˵һĶеnʽӰɡ
ţҪ˵Nļϣʽ

 nNH<n1>H<n>H<n2>H<n1>

ʽҪô
<For All>nNɡFor all n in N ˵ľǡnǡnɣ<For All>AllAд
һN̫һ
ǵģдNĻᱻһдN֪ܺⲻǼϡ
أ
ԪءʼϵʽҪʾеԪء˼д<For All>nNĻǴУѡһԪn˼
˼ǲѡĸ԰ɡѧѧѧдһӢĶѧġٵЦ˵
ѧġѧȷʵҲһ棬󲿷ֵʽᾭѹԼķʽ֣Ҫʽдʲôʱ򣬻ȽϺá
ʽŨ֭һһȥĻΣ𣿡
ˣǽʽд˵

H<1>1/1
H<2>(1/1)(1/2)
H<3>(1/1)(1/2)(1/3)
H<4>(1/1)(1/2)(1/3)(1/4)
H<5>(1/1)(1/2)(1/3)(1/4)(1/5)

˳¿ʱҪעӵĲ֣ҲHn1Hnࡡ

H<2>H<1>1/2
H<3>H<2>1/3
H<4>H<3>1/4
H<5>H<4>1/5
H<6>H<5>1/6

H<n1>H<n>ֵͻὥС͸ղŵٵ˵һ
ǵġ
H<1>H<2>H<3>H<4>H<5>ȻԽԽ󣬵ǡҲǡȴ𽥼١ÿ֮ҲԽԽ٣Ļ
һ£ӵԽԽ١Ĳ֣Ҹոдʽ֣š

еnH<n1>H<n>H<n2>H<n1>

ûȻӵԽԽ˵ʽдͺˣҲñȽ⣬Ȼкܶ˾ʽȸ⣬ܶ״дʽѶʽһԣֻҪʹõõܰԼ⣬ǳΪѶϢùߡ
õģҿĿǰʽдĻд𣿡

<For All>nNH<n1>H<n>H<n2>H<n1>

ţ˵
ٵƺܸˡ


825Ǵڵ

ôҪı
ף
ĿǰΪֹH<n><k=1n,1/k>nĻH<n>Ҳű󣬲ԽԽ٣һnʱH<n>Ҳ᲻ϵر𣿻˵ʹnϵH<n>Ҳ󵽳ĳأ ɡ
Ҳʽӻ᲻ϵرƿɣٵٽԼͷ˵

 (1/1)(1/2)(1/3)(1/4)(1/5)

ûǴſƬȻȻᷢչ⣬ҪĿǷɢʽдд

81
ʵΪRΪNʽǷ

 <For All>MR nNM<k=1n,1/k>

裿
費ƬyoǽExistsEдļǺš
ڡҲ˵For all M in R, n exists in N 
ٵٵķƯán existsáthere exists nҲûϵsuch thatĻ͸ˡ

For all M in R, there exists n in N such that M<k=1n,1/k>

ѧýҪô˵أ
ӲҪ˵Ļͻ仰

ʵMnʹM<k=1n,1/k>
Ȼ܆¡ǿܶٵٻش
(a)(b)ʽӣ֪ﲻһ𣿡ڱʼǱдʽ

<For All>MRnNM<k=1n,1/k>(a)
nN<For All>MRM<k=1n,1/k>(b)

΢е㳤Ϊ˼һ㣬ͼŰɡҼŲԵٵ˵

<For All>MR{nN[M<k=1n,1/k>]}(a)

nN[<For All>MR{M<k=1n,1/k>}](b)
([]ΪMԼΧ,{}ΪNԼΧ)
ӢдĻҼдӢġ
For all M in R, there exists n in N such thatM<k=1n,1/k>(a)
There exists n in N, such that for all M in RM<k=1n,1/k>(b)
ٵٽӢ˼Σ˼
ҾӦö˳Ҫɣ(a)ȾMȻnnʱM䣬(b)ȾnȻnѰM𣿡
ǵģ˼ˡ(a)һʼѡMȻMȥnΪڶȫMҵnѡMnͬҲûϵ(b)ҵnnʲôȫʵMʽ(b)£ѡMʱn䡣ε81˵(a)ûɣ
ǿûСٵ˵
Ҫ(a)(b)ĲǷǳѵģϸؿĻʽͿԺر֡
ȷҪֺѡ˵ʽĳֵMʲô
ʲôأٵ٣
ţܡܴ
žָоɣ֡һֱ󡻣ʵM֡M󡻻MֻҪ81MڵnͿ˵һֱȥǶκһMԵnͲ˵һֱȥ
ԭˡȻ˺ôһȦڽ˼Ūˡ
ţ𣿡
᲻һһ۶ѡѧ˵ҵġѧĵ㡻Ӳ١
ˣٵ٣͵ΪֹɣԱʦҲҪˣ͵ѧٴɡ
ãѧҡǳร
ûѧȤɣʽԣͲ˼ม
˵ܺѧһ𡭡ţǵģûҲҪѧˣ


83ֹݵֽ

죬ݵֽС
ѧǶͣ½Ųۿ
λŮƽ̨ǰֵࡣ
һλǲŮ׶ȣһλϲټŮӢӢǸšǿ糤ҲǸж꼶ѧȻҸ׶ͬѧ꣬ͬࡣ
׶ӢӢΪıϺƵɣڲϷͬʱҲף·ҪԽٵˡⲻܣעʱ򣬲֪ʱֽˣʲôʱ򽵵ģǷǳ˼ĸоֹ¥һ˽ĸоӵһľ޴ȴһһסԶ֣ǾȻøٵӣʵҾȡ
ҵλÿԿ׶ȿƶָ(ůָ)ţȷʵֻܿصߵĵҵĶȴֻ֡
תȻ󵭳ڲŵһ̣ڳ˱к׶ӢӢվ˸
Ȥ𣿡׶ҡ
治˼飬ֻ޵ټȴ޵˵롣
޴ɢһȤȴɢìܡ׶¶ЦƵ΢Ц
ƺòͬİ˶ɡ
ûò̵ͬİ˶ͬʱƽȻԽԽСڸŵͬʱС˵Ķĸоһ˵ĻбݵļޣҪˡ
ܰ˵ӢӢ˹˵ҪĺѣΪֻе׾ͲȤˣҪÿڲ壬Ҫ̫в˼ĸоҺ׶ȵָǰҴæ
ˣϣ´÷˹״Э׶ЦŽ顣
ʲôӰˣ´һ𵯰ɡӢӢһ߿ЦһԼĽҡ
׶ʳָ߸裬һȥ
ǳá

84Ħ

ݵĺΣ׶ȿŵ͵ɿǰ¡
㿴ľʦſƬ𣿡˵껰ѿƬ¡

 (׶ȵĿƬ)

(1)

(ףҵĲһ)
׶ȲҵĻشͼ˵ȥ
Ҫо(1)(1)޴ɢѾˣ֤ܿͿɡΪҲҪòͬʽԿҵ뷨ȡ
ҴȻ׶ȷɿ˵šԭʦθҺǲͬĿƬӡƺӦĿǰ˵ѧԭˣϲŮ׶ʵ⡣
˵õٵٽ𣿵ٵ٣Ů׸ôأȻǺóѧж൱Ķƺûʶһ㡣
һʼԽ̵ѧõ̬ٵٶԻģеͬʱҪ˼ĸо˵ٵٽܣΪϵۻһһ׵¥ݣŻٵ˵ҽܣƹʽһءز仯Ȼһһȷϡңұٵ˫۾һֱŵʱ򣬾ͻᡭ

ι׶Ƚҡ

ޱ׶ȶҡ
ˣȫû˵̫ˡ
Ͽεˡ
׶Եһ۾߻Լλ
ǳ㡣


85޴Ĺ

ڽͼڲӣԲʹͼңҺ͵ٵپڱݵĴѧ֡˸λýм㡣
˼
ٵصҵ£΢ٵһӣƮеζߴϰеĳѶࡣ
Ҿؿʼдʽ

81
ʵΪRΪNʽǷ

 MRnNM<k=1n,1/k>

ٵԱ߿š

H<8><k=18,1/k>
(1/1)(1/2)(1/3)(1/4)(1/5)(1/6)(1/7)(1/8)
(1/1)((1/2))((1/3)(1/4))((1/5)(1/6)(1/7)(1/8))---Ϊ124һ
(1/1)((1/2))((1/4)(1/4))((1/8)(1/8)(1/8)(1/8))
(1/1)((1/2)1)((1/4)2)((1/8)4)
(1/1)(1/2)(1/2)(1/2)
13/2

΢ϢȻм˲ʽӦɣΪ˷廯Ͳ㵽󣬶ֻ13/2ΪֹȻֻH<8>H<1>H<2>H<4>H<8>H<16>һķ

 H<1>10/2
 H<2>11/2
 H<4>12/2
 H<8>13/2
H<16>14/2
.
    .
    .

Ҫʽӹ廯ѣmΪ0ϵʽ

H<2<mη>>1m/2

ǲʽɣǵʽĻͲH<2<mη>>ֵ𣿡
ڵĿĲҪH<2<mη>>ȷֵҪH<2<mη>>ܴʲô̶ȡ뿴ʽӣmֵܴĻᷢʲô飿
֪֪һֱmĻ1m/2ͻһֱȥԡţòȺĻֻҪmH<2<mη>>Ҫʲô̶ȶû⣡
侲ͷʼúõؿMʱΪĳһnʹM<k=1n,1/k>
õģҶˣ۶ԶMֻҪmĻ񡭡

M1m/2

һҵmֻҪmΪ2Mϵҵm֮󣬾n2<mη>ҲmnnҪnɣ

M1m/2H<2<mη>>H<n><k=1n,1/k>

 û81Ľǡ

81
ʵΪRΪNʽ

<For All>MRnNM<k=1n,1/k>

ԭʽ淽㣬ȻȷֵȴԴСĵطơٵһ˵һơ
ҵһˣ<k=1n,1/k>һֱȥ
ѧ治˼飬1m/2ͿԽH<2<mη>>ȥΪҪƼ˲ʽﻹáԽԽС1/kӳɦ<k=1n,1/k>Ȼһֱȥĺܲ˼顣ٵٲϵصͷ

ţǾԿһֱȥ˵ʽ֣Ϊ˼򻯣޶ȫ0󡣡ұ˵ڱʼǱд
Բֺͦ<k=1n,a<k>>һ0a<k>0 (k123)ʹá

 <For All>MRnNM<k=1n,a<k>> 

ʱΪnޣԳƦ<k=1n,a<k>>Ϊ޴ɢǶ壬Ȼԡ

<k=1,a<k>>

ֳa<k>1/k״81Ϊڶˡ޴ɢԿԵõĽۡ

<k=1,1/k>޴ɢ

ٵһֱҵıʼǱ˼
ʲôֻҪһֱȥͻ᲻ϵرȥȻޡ
ףո˵ֵĻรΣ

82
ʵΪRΪN<For All>kNa<k>0ʽǷȻ

<For All>MRnNM<k=1n,a<k>>

Ҿ82Ϊa<k>һֱϵۼȥĻҲn󡭡Ҳűԣܻӵ<k=1n,a<k>>Mʱ򡣡
ţȻ˽뷨ٵ٣Ȼô˵е֣޴йม
ףвôӣҲM״𣿡
Ȼ˵a<k>һµʽӵĻΣ

a<k>1/2<kη>

ף
״ȫka<k>0Ǧ<k=1n,a<k>>ȴֹرΪ

<k=1n,a<k>><k=1n,1/2<kη>>

ͰanĶ壬д

1/2<1η>1/2<ƽ>1/2<nη>

Ϊ˷㣬1/2<0η>֮ټ

(1/2<0η>1/2<1η>1/2<ƽ>1/2<nη>)1/2<0η>

õȱе͹ʽˡ

(11/2<n1η>)/(11/2)1

ȥӣ1/2<n1η>һͿʽ

1/(11/2)1

֮Ǽ㡣

2()

Ǹ˼ļ1/(11/2)1Ľ2ɣ
ףģļӦ1

<k=1n,1/2<kη>>1

Ҳ˵  еnж󣬽1ϡۼ˶٣ȥἫȵؽӽ0Ժû취ۼӵ1ϣȻM1ʱnڣM1ĻnͲˣak Ϊ82Ĵ𰸻
82
ʵΪRΪN<For All>kNa<k>0ʽǱȻ

<For All>MRnNM<k=1n,a<k>>

ԭˣnʱ򣬻вֺͲ벢˵״ѧҲ󰡡
ȻҲʱȻԸղŵ֤ûʲôӰ졭
һ˲䣬ٵѧҵĿ˵
ҪúõȷϹ԰ɣѧ
˲ĳĬǿű˴Ц˳


86ڽ

ڷѧĽңҽסһ׼ȥ׶ȡ
׶ȣ֮ǰûú˵ҲԡǸ£ҶԦƺʵ̫ǹڦ(1)޴ɢĻ⡭
š
ǺѶԻˡ
׶۱ʣʼںڰд£
(s)Ķ壬ZETA

(s)<k=1,1/k<sη>>(Ķʽ)

׶ȼдȥ

(s)ʽ壬s1ǵͼõHarmonic SeriesһĸHдH<>

H<><k=1,1/k>(ͼĶʽ)

Ҳ˵s1ʽӾ͵ڵͼ
Һ͵١뵽 (1) һġ
ľʦҺ׶ȵͬԭHHarmonicĵһĸ
˵Ļ׶ȼ˵ȥ
ĲֺHnΪ

H<n><k=1n,1/k>(Ķʽ)

ҲǵnޣH<n>ͼH<>
ص׶÷۱дڰ

H<>lim<n,H<n>>(ͼĹϵ)

ΪH<n>nޣ޴ɢ

lim<n,H<n>>

˵ͼҲ޴ɢ

H<>

Ҳ˵(1)޴ɢ

(1)

Ϊʲô˵ͼ޴ɢ
׶Цһ£Ѿظƽģʽˡ
ڴȻ״̬˵дٵٵʽӣmΪ0ϵH<2<mη>>1m/2֤
û֤14Ͱķõͬķ׶˵

ͼ

(s)<k=1,1/k<sη>>(Ķʽ)
     H<><k=1,1/k>(ͼĶʽ)
     H<n><k=1n,1/k>(Ķʽ)

׶ʱ۾·ָӰָһLͣȻſ˵
㻹ǵɢҳָ𣿡
ţǵáӡַʽ⿪ġ
ôΣɢҳָķҲǶ

83
ӦĶS<eΪףx>ɢĺL(x)

      硡 ɢ
S<eΪףx> L(x)

Ҫȥˣɡ
׶Ͻϵķ۱ʻŪ֮ſڣͷ˵
ȸһ¡ȱǲͼѧɲֻʽѡ


87磬

ҹ
Լķ򿪱ʼǱ˼׶ȵ83
ɢУҳӦS<eΪףx>ĺ⡣
ǰָʱ򣬽˽De<xη>e<xη>릤E(x)E(x)Ӧ⣬ɹؽ΢ַʽַʽ˴˶Ӧ
ξʹӶӦ΢ַʽʼɡ
ϿS<eΪףx>΢֡

 f(x)S<eΪףx>
   ΢
f(x)1/x

΢֮1/xʣ΢ַʽ˼1/xҲдx<1η>Կá΢֮x<1η>֣׶ǰù΢DдĻͱʽӡ

DS<eΪףx>x<1η>΢ַʽ

ԴƣɢӦS<eΪףx>ĺL(x)Ĳַʽƽģ1ηȡΪݽ׳˵ģ1η

L(x)x<1εݽ׳>㺯L(x)Ĳַʽ

֮ǰ׶۵ʱֻǵݽ׳x<nεݽ׳>n0״ѡ

ݽ׳˵Ķ(nΪ)

x<nεݽ׳>(x0)(x1)(x2)(x(n1))

Ļʵؿn0״Ҫζx<nεݽ׳>

n4321ʱx<nεݽ׳>ͬʽӡ

x<4εݽ׳>(x0)(x1)(x2)(x3)
x<3εݽ׳>(x0)(x1)(x2)
x<2εݽ׳>(x0)(x1)
x<1εݽ׳>(x0)

ϸ۲Ļ֪ʡ

x<4εݽ׳>(x3)Ļõx<3εݽ׳>
x<3εݽ׳>(x2)Ļõx<2εݽ׳>
x<2εݽ׳>(x1)Ļõx<1εݽ׳>

ȻĻĹ

x<1εݽ׳>(x0)Ļõx<0εݽ׳>
x<0εݽ׳>(x1)Ļõx<1εݽ׳>
x<1εݽ׳>(x2)Ļõx<2εݽ׳>
x<2εݽ׳>(x3)Ļõx<3εݽ׳>

Ȼʽӡ

x<0εݽ׳>1
x<1εݽ׳>1/(x1)
x<2εݽ׳>1/((x1)(x2))
x<3εݽ׳>1/((x1)(x2)(x3))

ݽ׳˵Ķ(nΪ)

x<nεݽ׳>(x0)(x1)(x2)(x(n1))   n0ʱ
x<nεݽ׳>1       n0ʱ
x<nεݽ׳>1/((x1)(x2)(x3)(x(n)) n0ʱ

ŻصɣĿǽĲַʽ

L(x)x<1εݽ׳>

ԦĶL(x1)L(x)
ұx1Ķ1/(x1)Բַʽʽӡ

L(x1)L(x)1/(x1) L(x)Ĳַʽ

ҪܴL(x)ͺˡף
ף
L(x1)L(x)1/(x1) Ǻ͵ٵ֮ǰʽͬš

H<n1>H<n>1/(n1)H<n>ĵƹʽ

L(x)Ĳַʽ͵H<n>ĵƹʽȻȫһȻͶL(1)1ɣԵõּĹϵʽ

L(x)<k=1x,1/k>

ʹõıǷH<n>ͻģʽ

L(x)H<x>xΪ

ͽ83ˡ

83

L(x)<k=1x,1/k>
 H<x>

ȻĶӦϵ

Ĺϵ

 硡         ɢ
S<eΪףx> H<x><k=1x,1/k>

ڶ͵ʲôеĹûʲôо
һ£ۡ΢ֺͲ֡ʱ׶΢ᵽڡͷ֡飬Ϊ硻͡ɢ硻磬΢֡֡֡ͷ𡭡ãͼһ¡

磬

硻ɢ硻
 ΢D  Ӧ   ֦ 
   
 㡡  
 
 ֡ҡ Ӧ  ͷ֡ 

ţƯˣͼеġͷ֦Ҳ˵Ҫصµ硭ˣS<eΪףx>΢ֵĻ1/xҲ˵1/xֵĻͻɩS<eΪףx>ֵĵͷֵĵȫӦˣΪдɩS<eΪףx>ԿҪдɡ<x,1>(1/t)Ļͺˡ
ܼȥˡ

Ĺϵ
 
     硡                ɢ
S<eΪףx><x,1>(1/t)  H<x><k=1x,1/k>  

Ļdtд᲻ȽϺãɢ硭Ҫkˣٶk1Ļ˳Ӧ

<x,1>(1/t)dt Hx<k=1x,1/k>k

ԽԽ˳ˣܴ顣
ȱǲͼ
ء׶ôֱ˵ʹȽűʹˡ
ãǾ׶˵ͼͷֱʾ

[ͼֻƽֱϵy1/xͼ(1,1)㣬ͶӰϡ֮Ҳȡһ(x1/x)ͶӰxϡʱ(1,0)(1,1)(x,0)(x1/x)ĵͼΪ߽ΧɵĲͼεS<eΪףx><x,1>(1/t)]
    [ͼֻƽֱϵy1/xͼ(1,0)(2,0)(1,1)(2,1)ΧһΣ(2,0)(3,0)(2,1/2)(3,1/2)ΧһΡ(n,0)(n1,0)(n,1/n)(n1,1/(n1))ΧһΣεH<n>Ϊ<k=1n,1/k>]

มȷʵͼ֮󣬡硻롺ɢ硻ĺӦҲӾˡ˾ȡ


88֪Կףδ֪

ԵõĶ͡ɢĵǻӦġ
һڻؼҵ·ϣҺ͵ٵٲվҼ򵥵˵׶ȵҵĳɹ
ϸ룬ֻҪһ°ķ֤Ӧܷˣ㿴֤ ޴ɢʱ1248ÿ2<mη>ͻγһȺҲ˵ĸָķʽӣָܷķ֮ʵƾ͸ˡҪʱͼĻ̽⿪׶ȵ⣬ȫ׶ָһʹ
ԭٵζȻ˵ȴͻȻͣ½Ųֳһͷɥģ
ѧȻ˵Ҳо⡻ǽȫ޷ҳȤĶȫѧҵģҵѧȻǲС
มҲͣ˽Ų˵
ٵϣԼ˼ɣǺҪģʹôȴʲôҲûҲһΪŬ˼ˣԲҵĻҪһ㡣
ٵ˵
ҪʽʲôǷǳõһ£һʽֹͣ˼˷ǳ࣬˼ʽ֮ǰȫȥȻѵʽͲ׶ǾȫҲӦҪ롺Ϊֹ֪ģ֮Ҳ֪ġ˵û취ʱͻֹͣ⡢ֹͣ˼Żҽ˵ѧûжôԺһӡΪûԲɡҲѧѧʱԸսĵٵٷǳΰ󡣡
ǡѧʱȻ⣬ȴû취ֵزҪôأҪ￪ʼ˼أҾ÷ǳ˼顣
ʱ򣬾Ҳʲô·֣ԹȥĶ⵱ɻϿʱϰ⡢ԼĿ⡢дķ۵ĽⷨЩΪֱܷھԭ
ҼǰߣٵٸҵԱߣҼ˵
ʱ̬ʹòʽʽĴСһζʽһϵõ𰸣񡺴֪жϣ𰸻󣬵ǻǸС֮ġʹԼĿǰʹõķؽӽ𰸣һһȫ˽⣬֪ĵطȶШӣƶʯҲ֪Կ״δ֪š
ٵٵзâ
ٵ٣ѧϰʱԼжۻԭˡĸоɣԼ뵽ҲνĶ֤᡺̫ˡĸоҲǺҪľ顣
šţҶѧӢĵʱӢĵĸ˵ķҲϣԼķԷһѧѧĻ֮󡭡رлҡġ
һ˵һ߷Ųһõĵٵڻؼҵ·ߡ

ݵĳĬֻǰ
ˣҪҪȥ̨
סѧȥ̨𣿡ٵʳָָԼıӡ
ҴӶõѵдȯ˵ֵȥ㲻ϲ𣿡
ҺϲҪȥۡۡѧҺøˣǲλѧ梡Ǹ׶ѧ梡
˵Ҳǡٵ㲻Ļ
ᣡ᲻㣡ҾԻȥģ


89ֻĻ

ֻ˵Ļջ᲻ٺܶأΪ̫ŻȽаɣǵһֻ˵磬Ͳ鷳ֻǵ޵ʱҲ鷳ɣʱߣֻʣ˵ĻͲ𣿲ԻᷢȻˣǿʼи˳֣Ļڷḻ仯ͬʱ˵Ҳ˷ա
ʲô׶ҡ
ֻ˵Ļô
มѧʼǱ£Ǿ˵˵ֻĻɡ
׶һؽҵıʼǱ߿ʼдʽ

891ۻ

Ҵͷʼ˵˼ʽӵĻ׶˵ĬĬ

(2<0η>2<1η>2<ƽ>)(3<0η>3<1η>3<ƽ>)

޴ɢԳΪʽϵĻͷļչ۲졣

2<0η>3<0η>2<0η>3<1η>2<1η>3<0η>2<0η>3<ƽ>2<1η>3<1η>2<ƽ>3<0η>

ָĺĻܿɡ

(2<0η>3<0η>)(2<0η>3<1η>2<1η>3<0η>)(2<0η>3<ƽ>2<1η>3<1η>2<ƽ>3<0η>)

ҲĶغ֡

 <n=0,<k=0n,2<kη>3<nkη>>>

ҿʽӵչͷſ˵
׶ȣۻɣĦ<n=0,>n012ĹӣȻĦ<n=k,>Ӧnгָĺ͡Ҳǽ23ָƽ֡
ƽ֣ţҲ˵ֻӵ23ضֻгһΣΪ23ָ֣0ϵҲֹֻһζѡ
ԭˣȷһش
Ȼֻӵ23ӣҲ1Ų䡣


892ȱȼ

׶ȼ˵ξ˼ĻΪQ<2>

Q<2>(1/2<0η>1/2<1η>1/2<ƽ>)(1/3<0η>1/3<1η>1/3<ƽ>)

ղǸ޴ɢʽϵĻʽȴΪQ<2>ȴ˼ĹϵõȱȼʽʽĻQ2Ļʽ

Q<2>(1/2<0η>1/2<1η>1/2<ƽ>)(1/3<0η>1/3<1η>1/3<ƽ>)
(1/(11/2))(1/(11/3))ʽ

˵ȥԽQ<2>ͷչQ<2>ͻɡ͵ʽȻĸͻָղŵ2<kη>3<nkη>

Q<2>(1/2<0η>1/2<1η>1/2<ƽ>)(1/3<0η>1/3<1η>1/3<ƽ>)
1/(2<0η>3<0η>)(1/(2<0η>3<1η>)(1/2<1η>3<0η>))((1/2<0η>3<ƽ>)(1/2<1η>3<1η>)(1/2<ƽ>3<0η>))
<n=0,<k=0n,1/(2<kη>3<nkη>)>>͵ʽ

Q<2>Ҳ˵ĵʽ׶˵

(1/(11/2))(1/(11/3))<n=0,<k=0n,1/(2<kη>3<nkη>)>>

ǻұǺͰ˵

893ֽΨһ

ô衺ֻ23Ȼеһֻ ķĸ2<kη>3<nkη>һΡ 
ף׶ȣ2<kη>3<nkη>ܱȫɣ1Ҳֻа23ѣ5710֮Ķ֡˵
Բż衺ֻ23ֻ23Ͳ5710֮ˣ˵ʲô𣿡ش
˵ӷֽΨһ԰ɣ1Ψһʾ˵ֻ23ĻͲ57ֻĻ⻹ǵֹͣɡܾòʲô
֪ˣô˵Ļˣֻ2ԭֻ2Ǽϵֻm׶˵
⡭Ծ˵ˣ2mһĻͻ޸ˡ׶ȵҪ˵ʲô
Ǽ衺޸㻹û𣿡
׶ȵı飬ͻȻ뵽ˡ
֤𣿡


894޵֤

֤֤֮һ֮ܶǡҪ֤⣬䵼ìܡԷ֤ԴеֵҲ൱ࡣ

֤
裺֤ 
 
ì
  
 Ϊ
   
 ۣΪ 

ڿʼҪʹ÷֤֤ڡ
·ǰҽʦ㽫չԡ
׶ȣ˵֤ޣҪŷµķɣΪ޸ȫټ1
˵һʱ׶Ƚָ쵽ҵǰҡΣҪұ˵ȥ
Ϊ޸׶ȸɴؼ˵ȥ
ĸmȫС˳ԴСɡ

p<1>p<2>p<3>p<m>

3p<1>2p<2>3p<3>5ร˼޺Q<m>޻

 
Q<m>(1/2<0η>1/2<1η>1/2<ƽ>)(1/3<0η>1/3<1η>1/3<ƽ>)(1/p<m><0η>1/p<m><1η>1/p<m><ƽ>)
<k=1m,1/p<k><0η>1/p<k><1η>1/p<k><ƽ>>ʽ
  <k=1m,1/(11/p<k>)>
򵥵˵ǽղŵQ<2>еmȻΪm޵ֵˣҲ޵ġ
ҿʽ˼
šԭˣûp<k>2ϣԵȱȼ1/p<k><0η>1/p<k><1η>1/p<k><ƽ>1/(11/p<k>)ҲǻΪ޵ֵ
ûԴڿʼøȤ
˵껰׶ϸϴ
ղŶ23ͬƵmϣҲ޵ǰ¾չʽıַʽ˵βƽ֡mƽ֡

Q<m>(1/2<0η>1/2<1η>1/2<ƽ>)(1/3<0η>1/3<1η>1/3<ƽ>)(1/p<m><0η>1/p<m><1η>1/p<m><ƽ>)
(1/(2<0η>3<0η>5<0η>p<m><0η>))((1/(2<1η>3<0η>5<0η>p<m><0η>)(1/(2<0η>3<0η>5<0η>p<m><1η>))---------ָͷ
<k=0,<,1/(2<r<1>η>3<r<2>η>5<r<3>η>p<m><r<m>η>)>>͵ʽ

 ʽӡ׶˵
⡢ʽҲ̫رڲĦʲôûд
ȻʲôûдڲĦr<1>r<2>r<m>nȡr1r2r<m>Ϊȫܺ͡
ǡָ֮Ϊnȫϡ׶ȣ
û򵥵˵Q<m>1/(Ļ)ĺͣp<k>ָnʾָ֮͵ȫΪnȡ1/(Ļ)ĺ͡עĸҲǡĻĲ֣

 2<r<1>η>3<r<2>η>5<r<3>η>p<m><r<m>η>

ӷ֤ļ裬ϵֻmӷֽΨһԿ֪ȫ2<r<1>η>3<r<2>η>5<r<3>η>p<m><r<m>η>ΨһķֽⷨҲ˵Q<m>չĸ1/(Ļ)ķĸУеȻֻһΡ
š͸ոյ23һ
ĸСеȻֻһΡҲʽ˼

Q<m>1/11/21/31/4

ǵͼ
ڷˡ
Q<m>ޣܺȴᷢɢ
ûȱȼ֪Q<m>޵ġ׶ӭӶ˵ȥ

Q<m><k=1m,1/(11/p<k>)>(޵ֵ)

ȻQ<m>ֵڵͼ<k=1,1/k>

Q<m><k=1,1/k>(ͼ)

Ҳǻγʽ
<k=1m,1/(11/p<k>)<k=1,1/k>
ɷ֤ļõΪ޸ұǴӵͼõ޴ɢʽìܡ
˵
ӷ֤ļ衺Ϊ޸ìܣ˼ΪԽΪ棬ʾڡQuod Erat Demonstrandum֤ˡ
׶ʳָԡ
ã͵˸һˡ

ͼķɢȻ֤й˳Ծ൱صı
֤ǴӱΪһҲ˺ӥ˷һʦġ
ǵʦǡ
ʮΰѧҡ¡
׶˵

ͼ
 裺޸ 
   
ʽ Qm͵ʽ
   
    	   
   
ֵQm޴ɢ
   
 Ϊ
   
 ۣ޸ 


810̨


̨Сĸĸٵڵλ£Բһ̨״ĺɫͶӰ
ѧһ̨еţ÷ǳรٺ١ٵԼͷ
һˣͶӰһƬƻ辰ɫ̫£һһŵظ֣ҹղúССĹ㡣
Ư
Աߵĵٵٷ̾ȷʵ൱Ư

ڿʼǷ򱱼ɡ

˵ԱֹͣͬʱĻǳһתڿеĴҲŲؽӲǺܿ͵˱㡣
⣡֪ﴫСӵĽк
ϸ΢ĹâϵϣγһƬĻĲλཻϽΧ룬ۿҲڹ֮С
κ硢κʱ룬ٵ˵˱㣻ңԶ硢ңԶʱ䣬һ棬ȴ޵ǿա
ʱ

ҵࡸͨһ

ҵܵٵٵ
סҵ⡢忿ҵϣϴζøŨˡ
ٵ١
˵ԱӱԿЩбȻҵĶţȴﲻҵԺ
Ⱥ˸ҵȴֻߵĵٵӰ־ͻ˻ĵٵ١õĵٵ١ĵٵ١ɰʵףȴǷ¼򵥴ĵٵ٣רעһһ⡢ĵٵ١
ĵٵ٣ҡ
Ѿʲôˡ
ʹ޷ȫغϣһ£ʮʹỨʱ䡭ʹƹʽһн
ǹ޵ʱ⣬֪ϸ΢ޣǽΪ֪Ϊߣûг򣬵ԡ

ʱţյļ񱻴ɢʧ˵ԱȵǴʵ

λкúݵóһ

ȺǱ׹ɣղŷֵĻ˽ƶýĻ
ӻйĹۿǲȴɿƵؿԡ׼ҶԻصԼ
ǡ
Ȼٵץţڱ㣻ңԶ硢֮¡
šҪôȽϺأؿ

ף

ٵٿ˯ˡ
һ˯úܳ

ҡıʼǱ

ֺ͡<k=1n,a<k>>a<1>a<2>a<3>a<n>
<k=1,a<k>>a<1>a<2>a<3>
H<n><k=1n,1/k> 
ͼH<><k=1,1/k>(1/1)(1/2)(1/3)
  (s)<k=1,1/k<sη>>
ͼ (1)<k=1,1/k>
 (s)<p,f(k),1/(11/p<sη>)>
















9¡̩չʽ뱴


ｫ·ֿ
̽ܺ͡
׼ý޵ķ
Щ߱޷һ̽ʡ
[25]

91ͼ
911ſƬ
ѧˣ
ǳĵٵңϵĿƬҵ֣΢
ٵ٣ͼңҪϸ
ǵġԲ𡣡عͷ˼ؿ˿Χ
һͼңһķѧһĻŮٵ١
ȷʵͼֻǡֶ̫Աʦ鷳ˡ
šѧĿƬϵſƬһݸһţȻ˵ҵġһſƬǰ
ףٵҲõľʦĿƬ𣿡
ٺٺ٣ǰҸľʦ˵ѧѧʦ˿Ƭң˵һҵģһŸѧҽʲม
ٵ¶Цݡ
ҵĿƬдʽӡ

(ҵĿƬ)

 <k=1,1/k<ƽ>>

ٵٵĿƬǡ

(ٵٵĿƬ)

sin x<k=0,a<k>x<kη>>

ѧҵĿƬǡо⡻ɣָĵٵԱºʡ
ûо⡻ſƬΪ㣬Լ⣬ɵ˼ľʦʱǡ
ٵٽս˫ŵĿƬӦ˼ʽе˼
Ǹѧsin x<k=0,a<k>x<kη>>ַʽôⶼⲻ
ٵ٣ⲻx⣬Ҳ˵ʽӲǷʽЦŻش
ǡʽ
ţⲻǷʽǺʽ취ſƬϵʽӱɺʽҲҪxҪa<0>a<1>a<2>⡣
ѧȽһ㿪ͷʵʽʱһԼ͵ġֻҪһʼĲ־ͺá
ٵ˵껰ץ͸¥ݰһһ¥ݴŻһֱȥ
ֱ޵ԶΪֹ


912ʽ

Կ趨ɡұ˵ڵٵٵĿƬд⡣

91
躯sin xչݼʱ֮a<k>

sin x<k=0,a<k>x<kη>>

ݼǡ
ݼſƬұߵʽʽǡ磬֪xĶζʽɣ
𣿡ٵٴ򿪱ʼǱ

ax<ƽ>bxcζʽ()

ǵģ˵ȷټa0ȻĻa0b0ͲǶζһζʽˣȥԿ
õġ
	ϻش𲢶дڱʼǱϣ洿档

ax<ƽ>bxcζʽ(a0)

Ǹѧʽдܾú֡

ax<޴η>bx<ޣ1η>cx<ޣ2η>ʽ()

ԭˣٵٻд
Բԣ̫鷳ˣٵ٣ʽҪӴСʼдȻָּϡ޵״޴ηġޡɾں()֣ȽʽӾͺ

a<0>a<1>xa<2>x<ƽ>ζʽ(a<2>0)
a<0>a<1>xa<2>x<ƽ> ʽ(ݼ)

ԭˡдxָȽСĲ֣˵Ҳû˵Ϊʲôabcóa<0>a<1>a<2>أ
Ϊϵʹabczͱʾֻܴ025ηѣϾӢĸֻ26ҡѾxˣϵͲxa<k>һʹkڹ廯ʱҲȽףΪγɹ廯ô91ʽȥдд

sin xa<0>a<1>xa<2>x<ƽ>a<k>x<kη>

a<k>𣿡
ûûֻǽղ91ġƾдǽsin xı仯ֶΣa<k>⣬ҳa<0>a<1>a<2>ʵֵ
֪ʵʵֵa<0>a<1>a<2>ȫʵֵ
ûȫǺͼĻͻߣνߣͼҳa<0>ұ߻ͼ˵

 [ͼysin x]

ٵ٣ͼ뿴a<0>ʲô˵ֵ𣿡
ףҲ𣿡
ԿԣŬ뿴
ٵؿʽͼʼѰa<0>ֵ

sin xa<0>a<1>xa<2>x<ƽ>a<3>x<>

ٵٵı൱ḻĵʱŵʱ˼ʱ뷨ֱӷӦϣԽԽԼҲ֮衣
š۾ǵٵٵע̱갡תͫסŵĶҲкܲĸоҪǣֱʵԸΪһеĸҲû壬ٵپǵٵٰ
һ˵̧ͷ
ѧܼ򵥣֪ˣ0a<0>0
ûΪʲôأ
ͼ֪sin0ֵ0ˣΪͼͨx0y0ĵ㣬Ҳ˵x0ʽa<0>a<1>xa<2>x<ƽ>ӦҪ0sin0ȻҲΪx0ұߵʽֻʣa<0>a<0>֮ϵx0ʣa<0>a<0>ֵ0
ȷ𰸣̫
ԲҪϸ԰ɣ
ҲǣΪ0Ļͻ1ˡ


(֮غ0ͱɡ0!ѧǶ0!1)
ɣ֪a<0>ֵ𣿡
    Լa<0>0ȷ𰸵ĵٵ٣ٶĴ۾ʽҿʼ㡣
    ţŮٵڱҪʱļҲֲܿҲɫ֮һɡ
ٵٿʼ91ս
ǿʼҵĿƬ<k=1,1/k<ƽ>>ȴ򿪱ʼǱԶǦʣȡվ̬ʼ

ͼңǰؿʼù


92ѧϰ

ڻؼҵ·ϣҺ͵ٵ۵סլСվǰһŵٵٵĽŲߡ
sin xݼ˼ˣ

sin xa<0>a<1>xa<2>x<ƽ>a<3>x<>

Ϊ֪x0õa<0>0x/2xм㿴ϾҶsin˽ֻsin/21sinУ0֮ѡ
ʳָһС˵ΡΡΡһ߻ߡ
ԭˡҲ΢Ц
֪sin/21ҪġұߵݼԦ/2xֵȴ֪Һܴۣ
Ҫʾ𣿡
õġ
ٵ֪оǿ𣿡
𣿡ٵٱۣڳĶ
оǿ֮һ΢֡
΢֡һûѧţ˵ԼҲȤ
ٵⷽȽϱ𣿡

ͼݻкܶܺõ飬ѧϰõĲο鵽ר鼮ѡѧУʦѧϰϺҪǴͷβѧУʦ̫ˣȤĻ

ƺе㲻֪룬˵̫
ٵϲӢɡ鼮ɣ
ǰҳpaperback
֪ĵʱʦ𣿡
ᣬһԼֵ䣬ȵϿβѧΪȶ¡ѧҪ˵ľɣ
ǵģΪϲȥѧûбҪȴʦҲõȵϿΣԼ顢ԼĶѧø㡣
ȷڶӢ鼮ʱԼǰڴҪıأֻǲ鵥֣Ҳڴǵ֮ͬģԭѧҲҪѧϰҲǵȻġΪϿλûϵԺѧ̫õĸо
ˣո˵
Where were we
ף
ѧǵBeansһɡ
ڵٵ᰸ûκεֿ


93Beans


931΢ֵĹ

Ѿǵڼκ͵ٵһվǰĿBeansأʱѾϰ߲һˣΪʲôҪ˵ΪʲôĻΪŲĶʽº̯ʼǱ
￪ʼǲǺ΢ֺͶʽ΢֣΢һ㣬ѵĵطһ˵΢ֵĹ򡻵Ĺؼ
ûϵһ͵ģٵս˫ȭ
sin xݼ֡

sin xa<0>a<1>xa<2>x<ƽ>a<3>x<>

sin xȻؾʽ֣뾭֤ʱܣڵĿǽ⿪a<k> a<0>a<1>a<2>ʲôУsin xֽa<k>УǺݼչΪֹ𣿡
ٵصͷ
Ȼa<0>ֵڸղѾٵx0ҵˣΪ0ʽ

a<0>0

ٵٿ֮΢΢ͷܺãȥɡ
㻹֪΢֣ûʱ䣬û취΢ֵĶ忪ʼ˵ԾȽ΢ֵһּĹɣ΢ֵɡӺļ㡻ȻҲǴġ
Ӻļ㡻𣿡
ûf(x)΢ֵĻõһΪf(x)ĵf(x)ĵдf(x)ȻҲдʽf(x)Ƚϳʹá

f(x)f(x)
 ΢
f(x) f(x)ĵ

гһЩ΢ֵġ΢ֵĹ򡻣ѧ΢ֵĶ壬Щ򶼿ɶ֤

΢ֵĹ(1)΢ֻ0

(a)0

΢ֵĹ(2)x<nη>΢ֻnx<n1η>

(x<nη>)nx<n1η>(ָ½)

΢ֵĹ(3)sin x΢ֻcos x

(sin x)cos x

Щ΢ֵĹǰa prioriΪgivenɡٵ˵
ףa prioriΪgiven(֮a prioriﵥʣʾģ򼰹ģġ)
ѡ΢ֵĹ򡻵һʼĹ߰ɡٵٸĿ˵
ţŽʽӶx΢֡ڱʼǱдʽӡ

sin xa<0>a<1>xa<2>x<ƽ>a<3>x<>a<4>x<4η>

(sin x)(a<0>a<1>xa<2>x<ƽ>a<3>x<>a<4>x<4η>)

΢ֵĽʽ𣬵ٵ٣

cos xa<1>2a<2>x3a<>x<ƽ>4a<4>x<>

˿΢ֵĹʽӡ
šǡ΢ֵĹ(3)ɣsin x΢ֻcos xȻұ߽΢ֵĹ(2)ڸ
ûȻӦҪ֤΢ӵݼϡ
Ϊʲôa<0>ʧˣ
Ϊa<0>x޹صĳá΢ֵĹ(1)ȥ΢ֻ0
Ҷˣѧ⡺΢ֵĹôµʽˡ

cos xa<1>2a<2>x3a<>x<ƽ>4a<4>x<>

932΢

ôʽӣٵ֪a<1>ֵ𣿼翴ycos xͼӦþ֪ˡ

cos xa<1>2a<2>x3a<>x<ƽ>4a<4>x<>

 [ͼycos x]

ףѵ˵͸ղһ𣿽0cos xʽӾͺ˰ šӶ𣿡

cos xa<1>2a<2>03a<3>0<ƽ>4a<4>0<>
a<1>

Ȼͼеcos01ԡ𰸰ɣ

a<1>1

ûҵͷ
ٵٵϸֳЦݡ
ѧҿ֮ҪôˣҪcos x΢ְɣ
ûΪҪ(cos x)ļcos xõġ΢ֵĹ򡻡

΢ֵĹ(4)cos x΢ֻɣsin x

(cos x)sin x

Ļcos x΢֡

  cos xa<1>2a<2>x3a<3>x<ƽ>4a<4>x<>
(cos x)(a<1>2a<2>x3a<3>x<ƽ>4a<4>x<>)

ͻ̧ͷĵٵϷ쳱

sin x2a<2>6a<3>x12a<4>x<ƽ>

ţûҪϵǣʡ
a<2>֮ǰһx0ٵѸٵдڱʼǱϡ

sin x2a<2>6a<3>x12a<4>x<ƽ> õʽ
sin2a<2>x0
a<2>0 sin00ʽ

a<2>0ˣƺֵǿˣ״ԽԽáõģһ΢ֵĹʲô
Ѿˡ
Ҫsin xsin x΢ֵõ
û֮ǷȦ
Ȧ
sin x΢־ͱcos xcos x΢־ͱsin xеġ4ΪڵѭǺ΢ֵҶԵٵٽʡ

Ǻ΢
	      ΢
 sin x  cos x
΢֡΢
   cos xsin x
         ΢
֪ˣa<2>

sin x2a<2>6a<3>x12a<4>x<ƽ>ղŵõʽ
(sin x)(2a<2>6a<3>x12a<4>x<ƽ>)  ΢
cos x6a<3>24a<4>xá΢ֵĹ
 cos06a<3>x0
 a<3>-1/6 cos01ʽ
õģa<3>-1/6ˣa<4>
Եһ£ȻһһȥҲǿԣͳȫ忴ûɣ
ףû⣡


933sin x̩չʽ

ǺѾĿȣʼǱµһҳҸͷϵʾٵԼڱʼǱдʽ
Ҫsin xݼչղѾϵеa<0>a<1>a<2>a<3>ĸžҪͳдһsin xݼչɡ˵
õģɡ

sin xa<0>a<1>xa<2>x<ƽ>a<3>x<>a<4>x<4η>a<5>x<5η>

ţͺãˣxдx<1η>ɡ

sin xa<0>a<1>x<1η>a<2>x<ƽ>a<3>x<>a<4>x<4η>a<5>x<5η>

ͬʱ΢֣ڲҪϵԻʽǹؼ
ףѧҪ𣿡
ûҪ㣬ûȽҳԡԿҪرעⳣ

 sin xa<0>a<1>x<1η> a<2>x<ƽ>a<3>x<>a<4>x<4η>a<5>x<5η>
΢
 cos x1a<1>2a<2>x<1η>3a<3>x<ƽ>4a<4>x<>5a<5>x<4η>
΢
sin x21a<2>32a<3>x<1η>43a<4>x<ƽ>54a<5>x<>
΢
cos x321a<3>432a<4>x<1η>543a<5>x<ƽ>
΢
sin x4321a<4>5432a<5>x<1η>
΢
 cos x54321a<5>
΢
    .
	.
	.

õģ
ѧҵԡˣ54321Ļԭˣ΢ֵĹ(2)ġָ½ЧˣѾ˽йı仯
ûûԼдʽָоֻǿдҲǺҪģٵ١

ĵx0۲ʽӻôɡ
õģСʱǣţ۲ռһ۲ɡţΪsin00cos01

0a<0>
11a<1>
021a<2>
1321a<3>
04321a<4>
154321a<5>

֡ԡˣ
ţߵ1дɣ1ȽϺãҪa<k>ʽĸa<k>ߣ54321ǽ׳ˣд5!ݼչˣʽakд

sin xa<0>a<1>x<1η>a<2>x<ƽ>a<3>x<>

õģ0ùܣԾa<1>a<2>a<3>ãˣ
ţٵдݼչΪsin x̩չʽม

sin x̩չʽ

sin x(x<1η>/1!)(x<>/3!)(x<5η>/5!)(x<7η>/7!)

˵ǵٵǉչʽء



Ѽøӡ
ȷʵǺܸӣϸʽԶƵʱĺۼĸ1!3!5!Щ׳˾ǲ΢ָ½Ľ룭ŵĽûxżηⶼ0101ĻԼֵĹʽǡ
ԭˣûôѡɡ
ⲻý׳дĻͻγɺнеȤʽ

sin x(x/1!)((xxx)/3!)((xxxxx)/5!)((xxxxxxxx)/7!)

סƯдҲ𣿡;
Ȼû⣬ΪȤǷ⣬ԺֲܶͬдรҲx<ƽ>дxxԵʱдxxͲ̫ãôھͿԽƬϵ91ˡ
ףõģѾ˿Ƭˡɡ

91
躯sin xչݼʱ֮a<k>

sin x<k=0,a<k>x<kη>>

k4õa<k>ࡣ˵

91
a<k>0       k4õΪ0ʱ
a<k>1/k!  k4õΪ1ʱ
a<k>0       k4õΪ2ʱ
    a<k>1/k!  k4õΪ3ʱ
934ļ

˵΢˼һsin x̩չʽںдһsin x̩չʽ
Ǹдне̩չʽܾдд

sin xx/1(xxx)/(123)(xxxxx)/(12345)(xxxxxxxx)/(1234567)

ٵ٣ʽҲ޸ɵĺͣȡ޸Ĳֺ˼ｫxkΪֹĲֺs<k>(x)Ȼs<k>(x)Ҳxĺ

s<1>(x)x/1
s<3>(x)x/1(xxx)/(123)
s<5>(x)x/1(xxx)/(123)(xxxxx)/(12345)
s<7>(x)x/1(xxx)/(123)(xxxxx)/(12345)(xxxxxxxx)/(1234567)

Ҵӱóͼֽ
s<1>(x)s<3>(x)s<5>(x)s<7>(x)ͼҲǽys<k>(x)k1357棬Ϳطὥӽsin xҽͼ

 
 [ͼysin xys<1>(x)x/1ͬһƽֱϵ]
 [ͼysin xys<3>(x)x/1(xxx)/(123)ͬһƽֱϵ]
 [ͼysin xys<5>(x)x/1(xxx)/(123)(xxxxx)/(12345)ͬһƽֱϵ]
     [ͼysin xys<7>(x)x/1(xxx)/(123)(xxxxx)/(12345)(xxxxxxxx)/(1234567)ͬһƽֱϵ]

มԭˡѧҲsin xݼʲô˼Ȼá΢ֵĹ򡻵ʽӣǻ֡أĸоͼk֮sk(x)ӽsin xͶˣҺӺÿɰ
ǰ
ǡǸѧȻҲ̫˵Ǹsin xֻǸưɣֻǽĳsin xʾѣ̩չʽҲǽͬĺݼʽ֡͡sin x롺ݼ˵Ȼ൱һǺûиı䣬ԣݼʽ൱㡣˼Ҳ̫
᲻ᣬٵ٣ĺ˽ⱾʣҪоʱ̩ܽչʽʽչڱȽϷĶʽӳϽоղһ˼ĻǳãΪ޴Աرע㣬Ǹݼͻʮַ㣬˵쳲кͿʱõɺҲݼ
ܾúѧУϿʱȥעȽϸ΢ĵط˽ҪԶҪѧʲôʲôеһƬҡѧĻ෴ϸ΢ĵطͺ֮ԼһԹǶΪʲôʲôִȴס
šΪٵٵ

ٵٴҪ˵Ļ
ѧոȫ΢֡ݼ̩չʽչʽЩҲǵһһȴˣ̩չʽͨĶʽһоȻҪһܻСѵĺʽݼxk޺͵ķѾˡ
ٵٽȭͷǰ
Ҳǽѧ̹ҵ̩չʽѾȷʵˣҪоʱ̩͡չʽԿɡⶼѧĹ͡
ĿͻȻҵƿתϵͼֽٵٵϲ֪ΪȾһĨ̺졣
ԡԡȻҺܱǸ˷ѧıʱ䣬Һϲѧ˵ǳϲѧҡ
ٵٱ˵̧ͷȻ˵

һӶѧ̹ҵ̩չʽ


94Լ

ҹ
Լķ俴ŴӴľʦĿƬ趨ҵ¡

92
ֵ˵֮

<k=1,1/k<ƽ>>

Ƚдץסʽӵĸо

<k=1,1/k<ƽ>>1/1<ƽ>1/2<ƽ>1/3<ƽ>1/4<ƽ>1/5<ƽ>

Ȼһƺô׾ҵȼֵҲ˵Ǽ ǽ ĲֺԾn㣬춼עٵٵĿƬֻ㵽һ룬Ӳ㿴

<k=11,1/k<ƽ>>1/1<ƽ>  1
<k=12,1/k<ƽ>>11/2<ƽ> 1.25
<k=13,1/k<ƽ>>1.251/3<ƽ>1.3611
<k=14,1/k<ƽ>>1.36111/4<ƽ>1.423611
<k=15,1/k<ƽ>>1.4236111/5<ƽ>1.463611
<k=16,1/k<ƽ>>1.4636111/6<ƽ>1.491388
<k=17,1/k<ƽ>>1.4913881/7<ƽ>1.511797
<k=18,1/k<ƽ>>1.5117971/8<ƽ>1.527422
<k=19,1/k<ƽ>>1.5274221/9<ƽ>1.539767
<k=110,1/k<ƽ>>1.5397671/10<ƽ>1.549767

ţǲͼɡ
ף
Ҵ򿪱ȴûͼֽǷѧУ
ˣֺͿἱӣҲȷ֮ǰͼʱҲлɢļ
˵ĻʽӺ͵ͼʮơ

<k=1,1/k<ƽ>>ε92
<k=1,1/k<1η>>ͼ

һĵطֻһkָε921/k<ƽ>ĺͣkָ2ͼ1/k<1η>ĺͣkָ1
ָָ˵׶Ƚ̹ҹڦƺ£ҽĶʽٶдڱʼǱϡ

 (s)<k=1,1/k<sη>> (Ķʽ)

ʹ壬ͼԦ(1)ʾ

(1)<k=1,1/k<1η>> (ͼʾ)

92ҲԺʽдָ2Ԧ(2)ǡ

(2)<k=1,1/k<ƽ>>(92ʾ)

Ҳû취Ұ


95

֪ɣ
Ͻʱ׶ͻȻָ⡣
׶ҵͬͬѧѧ൱֣ѧУڵķΧԼϲ飬Լҳ⣬Ȼ⣬Ȼҵѧ㣬Ǹ׶ȱǲúԶҲҪԱУֻϣҲܿ硣
ѧáΰ£Ѿʼһһˣһվ鼮ǰţڵҴ󲿷ֶ޷ɡͬʱҲ뵽뵽֪ʶж㡣
ұò˽ԼѧأԼ£ֻšأƵطԼɣκ鿴ǴɣȻÿ춼ʽƺǱԶԶںͷ
ЩҲ޼£ٵһͣŬԼɡ
׶ȣǡnηʽn⡻𣿡
ţûϵnηʽn⣬ظͬ㡻
ó
ȻǸ˹ʦֵģ˾ȵʱŶʮ꣬òʿ֤òʿֻ֤Ǻܺġ
׶л໰ڿģʽˣ֮ǰƺͶһֱ죬һң̻صԼλ·ֳ໰ĲŮͰˡĸо
׶ȰڰǰʼϿΡ
ʵġֻҪ⸴ϵnηʽٻ1⡻ͺˣֻҪ1xʽֽnζʽڿʼҪ֤nηʽa<n>x<nη>a<n1>x<n1η>a<1>x<1η>a<0>0ٻһ⣬˼f(x) a<n>x<nη>a<n1>x<n1η>a<1>x<1η>a<0>ɣȻ麯ľֵ|f(x)|ܵССֵ0Ļͱʾн⣬֮ǰϰûɡ
׶ԷǳٶںڰдţҼʶ˹֤ұϿΡԼǷԸⲻ̫ȻоǶˣ֮ԼʵԼ֤Ȼڲ֤дſԣ׶ڵһʱܶ˵һֽ׶ˡ
˼ҿŴ׶дʽʽĽ˵ϿΡһ䣬ʼý⽫nζʽʽֽ⡣
дnηʽΪa<n>x<nη>a<n1>x<n1η>a<1>x<1η>a<0>0nΪ<1><2><n>Ļߵnζʽʽֽ⡣׶ȱ˵д

a<n>x<nη>a<n1>x<n1η>a<1>x<1η>a0a<n>(x<1>) (x<2>)(x<n>)

Ҳ˵ҪʽĽʽֱֽӵĹϵȻʽұߵһʼa<n>ͬʱǵߴx<nη>ϵͺ⡣ȳa<n>nεϵ1ȽϺ˼Ϊnζʽa<n>0˳a<n>û⡣
ʱҵƺڽҡ

ร봫еϵɫŮĻ棡
Ϊ꼶ѧöζͬѧǣɷ˵ؽٵٵٺͼֽݸҡ
ѧ˼Ҵ㣬ġ
ȻʣѧҸ˵ӡл𡭡е⣬Сϵɫʲô˼ӽԺҾͽ熪

иĻܸม׶ԾɶźڰдʽͷҲص˵
ʲôʱôЭˣҲô֡
ۡڰôʽ׶ѧд𣿡
ˣٵ֪ɣ
ǰĶ໰ŮҪԵٵٿʼϿΡˡ
׶ٵԳٶ˵ʽԼnηʽнϵĹϵ
ηʽax<ƽ>bxc0ĽΪ£ax<ƽ>bxca(x)(x)ͻҪʽĽ⣬ʽֱֽӵĹϵϵĹϵµʽӡ׶Ƚ˵

b/a
c/a

ͬηʽax<>bx<ƽ>cxd0ĽΪΪ£á

b/a£
c/a£¦ãæ
d/a¦

Ȼ廯nηʽa<n>x<nη>a<n1>x<n1η>a<1>x0ĽΪ<1><2><n>Ļ

a<n1>/a<n><1><2><n>
a<n2>/a<n><1><2><1><3><n1><n>
a<n3>/a<n><1><2><3><1><2><4><n2><n1><n>

(-1)<kη>a<nk>/a<n>(<1><2><n>Уk˵ܺ͡)
.
    .
    .
(-1)<kη>a<nn>/a<n><1><2><n>

ԭˣǡnηʽнϵĹϵɡ
ʱ𣬾ŮҲ˵ҵԴʽûˡ仰ҡҡλε߻һ꼶Ľҡ
ܿɰŮأ硣
˵׶Ȳһ䯺ָ۾֮们䲢¶Ķ䣬ָָڿռлߣҵȻȻͣߵ˲䡣
˵ߣҲϲͣ촽ǻһֱȥŷḻĻ񡭡
ưɡ˵
ף
ϴ˵ľʦǦƣӦ˵ˡ׶Ȱѿҿ

(׶ȵĿƬ)

(2)

Ȼ
֮ǰһʱ׶æƵĿƬҲӦͬľʦὫֲͬͬʽţٵٵĿƬͲˡ
Ѿ⿪׶ȣ
Ҫ˵⿪Ӧ˵ҼǵñĽõƬʱͻشˡ
⣿ǵá𰸣
ǰ(2)⡭˵֮ʦ¶Ц˵Ҫ𰸣ҪҴʽҳȤ⣬ҡ׶ʼ硣
šô𣿡
ǽʮͳѧȫ򵹵ĳ⣬ʦǳ֮ǰûκȷʦΪ⿪˱һٳ
һ£ôѵֻܽ⿪𣿡
ԡ
׶˵
ȻʮͳʱǸ⣬Ѿÿ춼ŵ
׶㻹Ŵ𰸰ɣ
ֻǿļѣѵôʦõƬ˼⣬뽫xΪzչķΧԿ
ม˵⡭԰ɣ(2)Ƿɢ𣿡
֪𣿡׶ȳԾؿң۾һ˲䷢â
ǲǣ˵ˣղҵʧԣһ룬ȲҪҡһæش
ڿƬдˡ⡹

92
ֵ˵֮

 <k=1,1/k<ƽ>>


96ͼ


961ٵٵĳ

ѧִ֡
һͼҡһķѧ뿪ʼļʱŮٵٱı˹
ôˣٵ٣
һֱٵ٣˿ʼ˼ԼӦҲҪʼˡ
Ǹ첻sin x̩չʽһؼ˼֮зˣsin xxı仯һֱ0รƩ硭
˵ŻĵٵóԼıʼǱ̯ҿ

sinУ0sin2У0sin3У0sinnnУ0

n123ʱsinnУ0
ǰȻش𣬲е㷳ˣⲻȻ𣿶ҡ
ٵ٣˿n0µҪ廯ĻҪд

sinnnУ0n012

ѽûûȷҲи
0ҲǣxύͼλϾͽˣ

 [ͼysin x]

Һһ˸˹ͷˡԲæʱ򻹴㡭
ϾĿõٵеɥֻڸ˵ʱɥʱҲֵúֱӣҲ˼æĿڣ
ǹĻ⣬뵽ʲôأ
ûʲô˵ġٵٹ۲ҵɫСĵ˵
Ȼ󡭡
ҡ
Եٵٵһ仰еޱ𾪡

Žsin xʽֽ⡣


ʲô

sin xʽֽ⡻ʲô˼
ǡΪҵܶsin x0x뽫x

sin x0

⿪ʽร

һػٵټ˵
׶ѧ㲻˵𣿡ҪʽĽʽֱֽӵĹϵ
ȷʵûҪsin xʽֽ⡻ϸ˼塣
ٵԳĬҽ˵
ղѧ˵һx0У2У3Уʱ򡭡

sin xx(x)(x)(x2)(x2)(x3)(x3)

Ϳʽֽ⡣
һûŪף𡭡ȷʵxnУĻ0û
ٵ٣֣Ϊsin xиļʽӡ

lim<x0,(sin x)/x>1

Ҳx0ʱӦû(sin x)/x1Ŷԣxǳӽ0ʱ ͻǳӽ1ٵٵʽУx0ͬ˲Ļͻʽ

(sin x)/x(x)(x)(x2)(x2)(x3)(x3)()

x0ʱʽߵļֵȻ1ҲΪұߵļֵ1Ժ⡣


962ȥδ

ٵҲ˼𣿡
ۣ

ͻȻӱǶһȫûз׶Ⱦվں档
ŵĵٵٰѱʼǱǦʺжˤ£ʡƤͳߵĵ
ǵģ׶ȣٵٲ˼⣬sin xġʽֽ⡻
ѧǸʲôٵٱ߼ʱʡ
õٵٿƬ˵ƽ͵⣬ҵĿƬԦ<k=1,1/k<ƽ>>ֵ׶ȵĿƬԦ(2)ֵ⣬Ȼֵָ¡
ٵҵ˵󣬱ֳһɫҲǵȻģϾԼȫû˼⡣
ʱ׶ȼٵٵµıʼǱȻʼġ
ม
ǸȻٵѱʼǱûȴ׶ȵѹס֡
㰡׶ȵû뿪ʼǱ˵̵ٵǉsin x̩չʽ𣿡ţԭˣҲǴľʦսдšһӶǣǡ
ԡԲ𣡡ٵһûرʼǱ
š׶ȱ۾ָһڶָ״̬ʱΧ˶ᱣֳĬע׶˼ʱһǵ
׶۾
Ǿʹsin x̩չʽʼɡ
һ˵껰ҵıʺͱʼǱʼдʽ

sin xsin x(x<1η>/1!)(x<>/3!)(x<5η>/5!)(x<7η>/7!)sin x̩չʽ

x0ͬxõʽǽ ԡ͡ʾ

(sin x)/x1(x<ƽ>/3!)(x<4η>/5!)(x<6η>/7!)x0ͬx

Ȼٵǉ˼ʽ

sin x0

ʽĽʾȼٶ׶ȼ˵

xnС(n0123)

sin xʽֽ⡻ǵٵǉ뷨ɣ
׶ͻȻβĶʷٵٵ˵ͷȻմ׶ûıʼǱǰǱдšһӶǣıʼǱԼ̩չʽĵٵ١
̲˳x0ʱ(sin x)/xļȻ1ҵʽֽȴСٵٲ˵
Ļ׶ϸֲı˵Ļsin xʽֽ⡻أ

sin xx(1x/)(1x/)(1x/2)(1x/2)(1x/3)(1x/3)

Һ͵ٵٻԿȻ󿴻׶дʽֽʽӣٵϴ򿪱ǰűʼǱʼ㡣
ţȷʵx0ʱȫ0ҵxnеʱΪ(1x/n)ʽڡҲ0Եx0У2Уʱ򣬻ʹsin x0ˡ
Ļ֮Ҳ˵
ң(sin x)/xʽӱʾx0ʱҲ(sin x)/x1
ڵٵٵıʼǱд¡

(sin x)/xx(1x/)(1x/)(1x/2)(1x/2)(1x/3)(1x/3)

ٵǉ׶ȴֳ
ٵǉŽղд(sin x)/xұʽټ򻯿
ٵٷˡš֮Ϳʼ˼
򻯡ĿԣΪĻƽĲ(1x/)(1x/)1<ƽ>x<ƽ>/<ƽ>ſԡٵһۼд¡

(sin x)/x(1x<ƽ>/<ƽ>)(1x<ƽ>/2<ƽ><ƽ>)(1x<ƽ>/3<ƽ><ƽ>)

ܽеأš׶֪ĻﲻΪ޷侲׶֪ʲôֵʲôزΪʲôҪ۱⣿ľʦսʲôȫҲ˽¡Ԥл˲Ľ
׶ȶ˼е˵ڵٵǉ á֣Ϊʽֽǽʽû֣⣬д̩չʽҲͬ á֣͡ԡ
׶ͣ˿֮˵ｫٵǉġġ͡ΪȰɡ

(sin x)/xʽ(sin x)/x͵ʽ
(1x<ƽ>/<ƽ>)(1x<ƽ>/2<ƽ><ƽ>)(1x<ƽ>/3<ƽ><ƽ>)1(x<ƽ>/3!)(x<4η>/5!)(x<6η>/7!)

д׶ȷ·͵㽫Կʽĵٵ٣ͬʱ˵ٵǉҲ÷˰ɡ
ٵǺش𣺡ҪҪʲô
׶ԭضſ˫֣ϸ΢˵ȽϿx<ƽ>ϵ

ҿʽ
Ƚϵ
˲㡣
Ƚϵ
ס
ѵ
̫ˡ
ҿ׶ȡ
׶ٵٿȥ
ٵ١
ףôסף
һı飬ƺû֡
ߵx<ƽ>ϵʲôٵǉ֪𣿡׶
֪޵Ļɡ
ٵǉʵչڰʽչ

(1x<ƽ>/<ƽ>)(1x<ƽ>/2<ƽ><ƽ>)(1x<ƽ>/3<ƽ><ƽ>)

ΪһѦУѶ԰ѡ

ax<ƽ>/1<ƽ><ƽ>b(1x<ƽ>/2<ƽ><ƽ>)c(1x<ƽ>/3<ƽ><ƽ>)d(1x<ƽ>/4<ƽ><ƽ>)

壬ʽͻд޻

(1ax<ƽ>)(1bx<ƽ>)(1cx<ƽ>)(1dx<ƽ>)

߿ʼչ

(1ax<ƽ>)(1bx<ƽ>)(1cx<ƽ>)(1dx<ƽ>)(עǰʽ)
(1(ab)x<ƽ>abx<4η>)(1cx<ƽ>)(1dx<ƽ>)(չעڵǰʽ)
(1(abc)x<ƽ>(abacbc)x<4η>abcx<6η>)(1cx<ƽ>)(1dx<ƽ>)
.
    .
    .


มʲôءٵǉ߿׶չʽߡ
ʵǽ˵ĽϵĹϵ֪x<ƽ>ϵĹ𣿡׶˵
Ӹղſʼ׶Ⱦֻ͵ٵ˵չʽӵٶҲƽǹǵҪ⡣
ãҶˣx<ƽ>ϵabcdɡ
û޻иʽx<ƽ>ϵ(abcd)޺(abcd)Ϊչx<ƽ>ϵȻصղšʽֽ⡻ʽ׶ȼ˵

 (sin x)/xʽ(sin x)/x͵ʽ
(1x<ƽ>/<ƽ>)(1x<ƽ>/2<ƽ><ƽ>)(1x<ƽ>/3<ƽ><ƽ>)1(x<ƽ>/3!)(x<4η>/5!)(x<6η>/7!)

׶ȵ˵ȥ
չʱx<ƽ>ϵұ߸ʽġx<ƽ>ϵ͡Ҳabcdɣ1/(1<ƽ><ƽ>)1/(2<ƽ><ƽ>)1/(3<ƽ><ƽ>)1/(4<ƽ><ƽ>)һ棬ұߵġx<ƽ>ϵܼ׶ǵĿǰΪֹʽߵx<ƽ>ϵȽϣĵʽͻ

1/(1<ƽ><ƽ>)1/(2<ƽ><ƽ>)1/(3<ƽ><ƽ>)1/(4<ƽ><ƽ>)1/3!

ٵȷϹ׶ȵĵʽ֮˵Ҫx<ƽ>ϵɡãҶˡ
û𣿵ٵǉ
ʲʲô°ٵһ˫۾ŵš
׶ȶԵٵǉ¶Цݣ·˵ôҲûϵԱʼǱٵ˵
ʽ֮󣬾ͱ

 1/(1<ƽ><ƽ>)1/(2<ƽ><ƽ>)1/(3<ƽ><ƽ>)1/(4<ƽ><ƽ>)1/6

߳Ԧ<ƽ>

 1/(1<ƽ>)1/(2<ƽ>)1/(3<ƽ>)1/(4<ƽ>)<ƽ>/6


ٵٴ˳ȻͼңҪ󺰵顣
⿪ˣĽ⿪ˣ⿪ˣ
ٵٿ׶ȣȻٿҡ
׶ȵͷӽ˵
⿪ˣѾ⿪ˣʮѧҵ⡪ѾýˣһĽⷨ
׶Ƚʽ¸д

<k=1,1/k<ƽ>><ƽ>/6

ȻҲддһʽ

(2)<ƽ>/6

õģ͸һˡ׶ͷʳָ¶ЦݣЦݡ
ΪΪʲôʲʲôʱ̫̫ˣ
ٵȻ֮С

92()

<k=1,1/k<ƽ>><ƽ>/6 


963޵ս

⿪ǵʦ¡ϵһ⿪ˣڹԪһ꣬ʦʮʱҲĵڶꡭ׶ϸ
ǷԽͰʱ䣬Ľⷨأʱǲֻʮꡭһǽĵڶꣿ
ǽ⿪𣿡ٵʡ
ǵģʦºü⿪ķһֽⷨ
֤һʱѾ㲻ˣǾ÷ǳȡٵٽȥ˵ҶԲ֪ͽ⿪ǳȣԭֻΪȻxnsin x0Ľ⣬˵Խsin x֡ҾǸܴķ֣ҲֻܵΪֹ׶ѧȴҵʽֽⷨһӾñȽx<ƽ>ϵķѱ⿪ˡ
Ȼ󣬻һ㡭ٵټ˵ҶԦ<k=1,1/k<ƽ>>ĺǦ<ƽ>/6𾪣Ϊʲôĵƽͻܳأ

ĬΪǶΪԲʦôͻȻֶе˼顣

˵Ϊʲôٵٵġʽֽ⡻вͨأʣxnС(n12)sin x/x0Ľ⡣

sin x/x(x)(x)(x2)(x2)(x3)(x3)

׶Ƚҵʡ
nȷʵsin x/x0Ľûʽֽ⡻̫߳Ҳ̫ˡϾֻxnǽĻֱӳCҲǷֽ֮ⷨһûΨһԡ

sin x/xC(x)(x)(x2)(x2)(x3)(x3)

šԭˣ׶ȣlim<x0,sin x/x>1޷ֻʽֽ⡻ְ
ûnζʽĻͿnεϵһ˵ߴηϵǹ̶ģԿϵʽΪ֪ϵ Ƕ٣û취ߴηϵγɹĹؼ(xn)ϵʽ(1x/(n))Ļϳlim<x0,sin x/x>1֮ǰϷЧ
׶۾˵
ϸ˵ղŵ֤ȻȫΪҳsin x0Ľ⣬ȡͼxĽxnУⲢxĽϣǲû۵Ŀԣʵʦ֮⻹֤ʽsin xݼչַ֤ǼƯģͬɱȽx<ƽ>ϵ(2)Ƚx<żη>ϵҲ(ż)

Ľⷨ

  ٵٵĿƬ   sin x0Ľ
 
sin x̩չʽ sin xʽֽ
 
sin x/x͵ʽ  Ƚx<ƽ>ϵ  sin x/xʽ

   (2)<k=1,1/k<ƽ>><ƽ>/6  
 
 ׶ҵĿƬ

ȻңΪѾ֪ʦⷨĹϵ
׶ȱ˵վ
Ȼú÷ʽĽ⽫sin xʽֽ⡻Ƿǳ˲Ȼлв̫ϽĲ֣Ҳжսľġ
׶ȽַŵȻŵĵٵͷϡ
ڶǵʦʦʾͬʱҲҪΪٵǉ֡
׶ȿʼ֣ҲվԵٵֻ֣˵Ȳɡ
׶ѧ㡭ѧ
ٵٽסߺգ۾գš
ͼңҪ
ЩѾҪˡ
ڡΪǵĻŮٵ֣

ĺͣ
11/2<nη>1/3<nη>1/3<nη>һ
ù̬׵չ֣
nΪżʱԽԲʦеİæʾ
ʵϣЩͶԦ<nη>ıһֱ
[25]



10¡ֲ


׵Ĵӵյ
С[8]


101ͼ


1011ֲ

һķѧ
ˡ׶ȱ˵߽ͼңƺõ˴ľʦ⡣
Һ͵ٵϵֽš

(ľʦõĿƬ)
Ϊ1Ԫ2Ԫ3Ԫ4ԪӲң֧nԪϷʽм֣ϵĸΪP<n>(֧Ϊnķֲ𣬷ֲĸP<n>Ϊnķֲ)

˵֧3ԪķС3ԪӲ1ö2ԪӲһö1ԪӲ1öԼ1ԪӲ3ö3ַʽP<3>3

101

P<9>

102

P<15>1000Ƿ

ֻ֧ķӦúܼ򵥰ɡһ꼶ѧáŮٵ˵
𣿡˵
ףνP<9>֧9Ԫ״ɣաʹ1ԪӲҵ״ʹ2ԪӲҵ״˳˼Ϳ˰ɣ
ٵ٣ûô򵥣ΪʹͬԼʹʹ1Ԫ״ҲҪǵʹ˼ö
ѧҲһֱǲСĵٵǉรöҺֻҪ侲ûˡٵƺš
һֱĻʧรùĽⷨȽϰȫ101P<9>Ȳᣬ102P<15>Ӧûɡǳ
ѧǳǶֻ֧15Ԫİɣ
ٵ٣Ȼֻ15ԪĻ̾ͻᱬ



һֱû˵׶ӣ汬Ч
Ȼͣ¶Ի
ٵǉǸ䣬ȥߵλãҶҪС˵˼
׶һ£ٵٺҶͷͬ⡣
֪˻ж

Ǿڷѧͼؿʼù


1012˼ʵ

(1234)仯ӲңЩӲ֧nԪ֧ķֲ⡣
ʹǰһȴСо˼ʵץоǺҪġ
n0ʱҲ֧Ϊ0״ֻС֧һַΪ1Ҳ˵P<0>1

P<0>1֧0Ԫķ1

n1ʱ򡭡ֻСʹ11ԪһַP<1>1

P<1>1֧1Ԫķ1

n2ʱ򡭡Сʹ12Ԫ롸ʹ21ԪķP<2>2

P<2>2֧2Ԫķ2

n3ʱ򡭡Сʹ13Ԫ롸ʹ12Ԫ11Ԫ롸ʹ31Ԫ3ַ
дȥ̫鷳ԡʹ12Ԫ11Ԫַ21֣Ҳд

21(2Ԫ1ö1Ԫһö)

n3ʱµ3ַ

33
 21
 111

ͬP<3>3

P<3>3֧3Ԫķ3

νP<3>Ȼǡ֧3ԪķҲ˵ɡ3ֲķԲűΪֲɡ
n4Ļ5֣һ5ֲţѾץ

44
 31
 22
 211
 1111

P<4>5֧4Ԫķ5

n5Ļ7֡

55
 41
 32
 311
 221
 2111
 11111

P<5>7֧5Ԫķ7

n𽥱ʱ΢һЩҪҳԾͲףǰ׶˵׷ֹ򡹣ӵ̫󣬾ͻѡ
ȥɣn611ֱַ

66
 51
 42
 411
 33
 321
 3111
 222
 2211
 21111
 111111

P<6>11֧6Ԫķ11

ţP<2>P<3>P<4>P<5>P<6>235711ǺйصĹ
P<7>13

77
 61
 52
 511
 43
 421
 4111
 331
 322
 3211
 31111
 2221
 22111
 211111
 1111111

P<7>15֧7Ԫķ15

P<7>15֣ϧ
Ҳʼˣȥn8n9ʱû𣿻᲻䵣ԡ
n8״

88
 71
 62
 611
 53
 521
 5111
 44
 431
 422
 4211
 41111
 332
 3311
 3221
 32111
 311111
 2222
 22211
 221111
 2111111
 11111111

P<8>22֧8Ԫķ22

ڵn9ˡ

99
 81
 72
 711
 63
 621
 6111
 54
 531
 522
 5211
 51111
 441
 432
 4311
 4221
 42111
 411111
 333
 3321
 33111
 3222
 32211
 321111
 3111111
 22221
 222111
 2211111
 21111111
 111111111

P<9>30֧9Ԫķ30

ţľʦ101ͽˣ9Ԫķ30֣9ķֲ30

102ôأӲP<15>ҲǡǳӦP<n>һɡ

ǱУʱ䡣

ʦˣѾô˰
ʦǻᶨʱֵĹԱʦ޷ж߷ɫ۾һߵͼݵʱ䵽ˡ
֮ȵѾ֪101ĴP<9>30ˣ֪102Ĵ𰸡

101

P<9>30


102ȥ·


1021쳲

վ·ϡ
ٵٰڳһҪȭĶ
ʲôʡ
쳲ơ
ʲô
Ҳѹѧ֪Ϊơ

Ǵҷǳϲѧ˼ǰѧ˴кķʽڼʱʹมΪƣԼʹԲͨǸһЩʱҲԴԷ֪ɡ
ٵǉ¶һӡ
Կɡ
ٵٽ쵽ҵıǰ棬ڶĴΡ
֪𣿡
֪ʲô
Ҫϸָȵ֣1123ࡡ
ٵ±һΣָȷ1123ӡȻ
Ȼ󿴵쳲ƵʱҪȳȭĲӦΪ1123һ5ָȳ쳲У쳲ơ
ม׶ȣڸղŵP<9>
ѧҪҰ
ҿ׶ȣָҲڸűȻ
ף׶Ҳƣ㵽ʲôеطʡ
쳲ơٵǉ5ԺҪô죿358𣿼쳲ƼȥĻȫ˵ֶࡡ׶˵
5ͣˡǾһȰɣұȳ1123ĻҪӦ5มһһ
׶Ц֡
˾ò˼ֲСѧ
ٵȻָô۾ңû취ֻֻӦ
塣
áлл
ŮҲǾ


1022

ߵ˴·ԣΪõ·Сǰյٵ١ҡ׶ȵ˳ųһǰٵٲϵתͷ˵ƺеΣգҶ׶վҵܾò̫ڡ
101ֵ˶102Ե˶ˡ׶˵
ٵתͷ˵Ѿ101ˣ102ʵдP<15>ʱ֪ǳʲô˼ͻȻˣȻд50֣ȻûнԲᳬ1000֡
ٵ٣ǰ濴ǰ棬ҪСǰĵ߸ˡ
ûģѧ˵P<9>29ְɣٵٰѱʼǱǴ򿪡

١8 ڣߡ ١2
ۣޡ ٣ڣޡ ١3
ܣݡ ٣ۣݡ ١2
١2ڣݡ١4ݡ٣ܡ2
١2ۣܡ٣ڡ2ܡ١3ڣ
١5ܡۡ3٣ڣۡ2
١3ۡ2 ڡ3ۡ١2ڡ2
١4ڣۡ١6ۡ٣ڡ
١3ڡ3١5ڡ2 ١7
١ᡡ ᡡ 

 ס29֣30
Ҫôʡ
͸㿴һ١3֧1ԪӲ3˼
รԭˣдࡣ
ûТڣۣܡ׶ԽҵļűʼǱ⣬ĳңͬʱĹ㡣
ѽȷ˺üΡʹٵٵͷײƣҸողǲѹ

վˡ
Ҿˣ׶һµٵٵͷ뿪·ߺǸպ෴
׶ѧˡһ쳲ġٵ죬ϵضָͷ
ʱ׶Ⱦָ֣ڿл裬ûзٶȣҲûлͷ

103Beans

ڵٵҪҪȥȱȺǾ˳վǰġBeansҶλϡ
ٵٽţ̼뿧ȺֻڷҲ׽裬ƺƽе㲻̫һŻ˵Ļ
Ҳϣѧܱøһ㣬ȻûҪֻǲУ࣬뵥P<15>P<15>Ǹǳ
̾һ
ѧûҵλء
ף
ףٵٵԽԽ죬Ҹղ˵𣿰ղŵĲ㣡ԣҲǲ㣡ǵġ
ֻǰȥⲻ쳲ơ
һ
ٵٵͷ
ѧ꼶ʱҲҹж꼶ʱѧУʱзɣǹڶƵ⡣ڷѧ˵ˡѧǳԽʱġʷѧоƣǲִĵѧǳԽʱġҲҵ˵ʮоƵȲöʮһеԵ£ȻѾȥˣѧȴԽʱţִˡҴѧĻܵ飬ûĵΪѧǿԳԽʱġ
ô˵Һз⡣
ѧʱѧ󶼻ᵽͼȥɣУ֮ΪҪӽѧҲʼȥͼ䡭ҾͼĽ俴飬ѧһֱæż㣬ȫûע⵽ңҲΪ궬һͼĽ
̧ͷ¶˼Цݡ
ûע⵽һֱΪû˻ͼֻһˡ
ҡ֪ܺѧͬһеʱĺܸˣ¶дŸѧ̫ˣҷǳϲҡٵ١˵ٵٺʱľúʲôɹѧҲȥ̨ܸѧ׶ѧһѧҷǳ֡
ô˵ȥ̨û
ҲдͷɥʱȻܺѧۣԼһȴʲôҾʧˡ
šܶ飬ҿ׶ʱҲһĸо
ҵλáѧҵλ𣿶ѧԣҿֻһıѧöѣѧǷҵλأֻżԸ̽ѧҲԡ
ҵĳ򡭡Ϊ˵ٵٶڵ򡭡
ҿ˵
ţʲôʱСϲ͵ٵ˵Һֱǰڡѧ֡Ӧһκʱ򶼿ԽѧţԼûиı䡭ҲֻԼһ˾ʲôеʱͼȻȤڸ֣Ϊܺѧˡҵڵٵٵλһֱڣ˵ҺҪѡ
ͣٵָ֡лллҷǳˡƺ˵̫ˣԵΪֹ
ѵĵ·ڿռ֮С
ԭˣÿڻؼҵ·ϣٵٻߵɡ
ϣһ
޵ʱ


104Լ

Լҡ
ʾ2359֣2359
˶˯ˣڷѧҸʱ̡
˶սʲôʽûȤȤʽ仯ܸ˵غ˵Ҳֻõһ䡸ѡ
Ǻģ׶ٵٿԺһ⡢һһ֣֪ʶս̸ֽ֮ⷨΪĹͨԡϲʱ䣬͹ʱͼȹʱͬʱֻһڼ㡭ԲԣǸʱǸطٵٻҲڰɡ
ľʦɣ˼ֲP<n>սǷɺP<15><1000102
νɺʹxĳݽȫϵһУڴ֮ǰҺ׶ʹɺ쳲뿨һεP<n>ǷҲɺҳһأֻҪҳһP<n>ġnıչʽѸٽ⿪102ˡ
ĿǰΪֹ֪P<n>

n   0123456 7 8 9
P<n>11235711152230

еɺΪP(x)P(x)Ķ£ͬʱɺĵĶ塣

P(x)P<0>x<0η>P<1>x<1η>P<2>x<ƽ>P<3>x<>P<4>x<4η>P<5>x<5η>

P<0>P<1>ֵʵʴ룬nεϵΪP<n>Իʽӡ

P(x)1x<0η>1x<1η>2x<ƽ>3x<>5x<4η>7x<5η>

Ϊ˲øһʽϵıxϵ̬ӵΪɺĸ׻
һɺġxıչʽ
쳲F<n>ʱõƹʽչʽxP(x)ϵת˻
C<n>ʱɺĻչʽҲᵽƽ֡Ȥ
ֲP<n>أҲɺ֮ͻħһ⿪бҪҳеıʡ
оֲɺɣҹ


1041Ϊѡ

ڷȦ˼ȻֵֺҪȥֻϵֱѡڱɡ֮ǰĽⷨҲ׶ȿСԵ˶˼˼ɡ

Ҵ򿪴ҹĿԶĹΪʲôһϲѧأѧʲôأ׶ô˵ģ

翵ж˵ѧıɡʦɵģ޴СĸԼоϣԲʵĦУλiȻĵeʦʼʹõ֣ʦڵʱ޷ɵĺϼһţڿ˹ϼһ

šδĳһµ

΢ɺ˼ԼǲнƵأ

ԲҼǲס
ԣҪҪ˼˼

ƺ͵ٵйֻأԼѡ˼Ҫ顺Цˣ˼ȻҪҲһ
͵ٵٵĶԻʽʱ򣬽(xy)<nη>չ󣬻ϵõٵʮ־ȣҲ֪()<n,k><,n,k>ͬ˼
(xy)Ϊnηʱn(xy)ʽиѡxyȻΪxy֮ͬ󣬾ѡķ
(xy)3չʱ򣬴xyĸʽ3ͻ8

 (<x>y)(<x>y)(<x>y)xxxx<>
 (<x>y)(<x>y)(x<y>)xxyx<ƽ>y
 (<x>y)(x<y>)(<x>y)xyxx<ƽ>y
 (<x>y)(x<y>)(x<y>)xyyxy<ƽ>
 (x<y>)(<x>y)(<x>y)yxxx<ƽ>y
 (x<y>)(<x>y)(x<y>)yxyxy<ƽ>
 (x<y>)(x<y>)(<x>y)yyxxy<ƽ>
 (x<y>)(x<y>)(x<y>)yyyy<>

Щȫͬ֮󣬾ͳΪչ

(xy)(xy)(xy)1x<>y<0η>3x<ƽ>y<1η>3x<1η>y<ƽ>1x<0η>y<>

ϵ1331Ƿѡ3x2x1x0xķҲ˵ϵ ʾĻΪʽ

(xy)(xy)(xy)()<3,3>x<>()<3,2>x<ƽ>y<1η>()<3,1>x<1η>y<ƽ>()<3,0>y<>

뵽Ժиֳٵŷߵͣ½Ų
ף
ʲôҪĵط⡣
ٵڸǰ档
ҪҪ˼֮
ѡķϵ
ѡķϵ
õٵٵķ෽ʽѡţйϵһͷֲɺйأʽ޺͵޻ͺˣ֪ˡ
֪˻ж׶ȵҵ
Ҽæʼ㣬޻ԲܳΪxıչʽӦÿԵõʽɺP(x)

ҹԼУҰؿʼù

103(趨)
ֲɺP(x)ʽP(x)


105ֽ

졣
ڷѧֽңӢӢҡ׶졣
ض˵ص͹
ӢӢһƽ̨ٵŸ±һ죬ͬǸж꼶Ȼͬѧ꣬ȴҺ׶Ȳͬ༶Ǹšǿ糤Ǹǳϲټ̵Ů
ţ͹׶΢Цؽֱڱһһֽգƺܺá
ٵǉû𣿲ǲӢӢ߶˵
ٵǉ
ûرҰɡһش
պʱ򣬱űʼǱĵٵٽֽҡ
ѧﰡһѧͼһȥء
(ⲻǺظں)ӢӢС˵
ŵ𣿡ٵǉǡ
ᣬٵ٣Ҳûʲô˵
ҪҸ˵𣿡
õĺõġˡ˵ٵãҪҪһѧɹ׶ȣҿдֲʽӰɡ
һP<n>ıչʽ𣿡׶ͻȻͣ½ŲϾʡ
ǣһP<n>ıչʽɺP(x)޻ʽһش
Ǿͺá׶ٶȻָЦݡ
ʹǰĺڰɡ
ߵֽǰ棬ƶʽڰ׼׶Ⱥ͵ٵҲҡ
ӢӢ˵ѽҪʼѧˡ󣬵ٵ־ͣ


1051ҵķ(ֲɺ)

Ϊ˽⿪102ҲȡķʽֲP<n>һΪȴɺP(x)ʼɺP(x)¡

P(x)P<0>x<0η>P<1>x<1η>P<2>x<ƽ>P<3>x<>P<4>x<4η>P<5>x<5η>

ԭĶ壬Լ趨ˡҳʽɺP(x)⵱103ڽ103֮ǰΪ˷˵104ǼӲöƵġķֲ

104ķֲ
1ԪӲҡ2ԪӲҡ3ԪӲҸ1ö3Ԫķм֡

104ѡӲΪ123ԪӲ3֣Ҹֻ1öҪ֧3ԪֻС1ԪӲ2ԪӲҡ롺3ԪӲҡ2ַǴ𰸡

104
ַ

Ȼ104˵ɺʹøӲ֧Ľб

ʹâ֧Ľ0Ԫ1Ԫ
ʹâ֧Ľ0Ԫ2Ԫ
ʹâ֧Ľ0Ԫ3Ԫ

ʽ˼ʹʽϵıxָԡ֧֣Ϊ˷⣬1дx<0η>

(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)

ԭˣȤ׶˵
ǰ΢Цظ
׶ѧ㣬ԭˡʲô˼ѧǰʲô˼أȫҪ˳˵ٵٿʼԹ˵꣬ӢӢĸپϲ֡
ȥɡ׶˵
ٵ٣ղŵʽҪҶ˵

(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)
    ٵĲ             ڵĲ             ۵Ĳ
չĻͺˣǽӲڸ֧״αָԸʾ֧Ŀԡ

(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)x<000η>
x<003η>
x<020η>
x<023η>
x<100η>
x<103η>
x<110η>
x<123η>

Ʃ˵x<120η>ָ120

1ʹâ֧ĽΪ1Ԫ
2ʹâ֧ĽΪ2Ԫ
0ʹâ֧ĽΪ0Ԫ

ѧһ£һǲx<120η>˼Ҫ1ö1ö0öĻָҪ120110ǱȽϺ𣿡רעʽϵĵٵı鷢ʡ
ǵģﲻǡkԪӲҵöҪákԪӲ֧Ľ롣
ҵĻΪkԪӲҵĹ׶ȡ׶顣
ѧ΢ˣȷչʽӺxָͿ֪â֧пԡ治˼飬ΪʲôһҪ(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)ʽأ
Ϊ(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)ʽӵչ֧͡пԡպһµĹϵչʱ²

-x<0η>x<1η>ѡһ
-x<0η>x<ƽ>ѡһ
-x<0η>x<>ѡһ

Ļͺ֧һ˼ʽ

-ѡâ֧Ľ
-ѡâ֧Ľ
-ѡâ֧Ľ͡

ԭˣҶˡΪȫϣʽչĳ˷ɡ١ٵƺܽˡ
Ҽ˵ȥ

չʽʽͬxkָСУҲͬ

(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)     ʽ
x<000η>x<003η>x<020η>x<023η> չ
x<100η>x<103η>x<110η>x<123η>
x<0η>x<>x<ƽ>x<5η>x<1η>x<4η>x<>x<6η>ָ
x<0η>x<1η>x<ƽ>2x<>x<4η>x<5η>x<6η>ͬָ˳

ٵ٣x<>ϵ2ม֪ʲô˼𣿡
ţϵ2ĻǳΪx<>2ʵϾ롭ԭˣ֪ˣx<>ϵ2˼Ǳʾ֧Ϊ3ʱ2ַ
ٵ˵һ¡ǰʹʽxĳݺͣȻx<nη>ϵǡ֧Ϊnĵķǡ֧Ϊnĵķʲôأ
֧ΪnĵķǷֲ
ǵģ104Ӳҵöƣľʦ101102ֵķֲͬȴƣʹʽxĳݺͣϵɡ֧ΪnĵķɺҲ˵(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)ʽӾǡƵķֲɺ

104ġƵķֲɺ

(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)

ԭˡ˵ɺͻ뵽ܸӵҲ(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)һСС޻ɺɺٵŵơ
ôҼ˵ȥ

򡡡򡡡

ôΪֹǡƵķֲڿʼҪӲƣ۵ķȻһȻѾ(x<0η>x<1η>)(x<0η>x<ƽ>)(x<0η>x<>)ġ޺͵޻ʽġ޺͵޻

(x<0η>x<1η>x<ƽ>x<>)ٵĹ׶
(x<0η>x<ƽ>x<4η>x<6η>)ڵĹ׶
(x<0η>x<>x<6η>x<9η>)۵Ĺ׶
(x<0η>x<4η>x<8η>x<12η>)  ܵĹ׶

(x<0kη>x<1kη>x<2kη>kx<3kη>) Ĺ׶


޺ͿӲҵöơ
޻Ӳҵ಻ơ
޺͵޻չʱ֧ʽĿԾͻһ֣ȡ֮ͬx<nη>Էx<nη>ϵnķֲΪx<nη>ϵ롺֧nԪķͬ
ϵΪֲʽݼҲд޺͵޻ΪֲɺP(x)Ϳд

P(x)(x<0η>x<1η>x<ƽ>x<>)
(x<0η>x<ƽ>x<4η>x<6η>)
(x<0η>x<>x<6η>x<9η>)
(x<0η>x<4η>x<8η>x<12η>)

(x<0kη>x<1kη>x<2kη>x<3kη>)


תӵ㣬ʽx趨0x1ΧʵȻõȱȼ͹ʽkԪӲҵĹ׶Ⱦͻķ

x<0kη>x<1kη>x<2kη>x<3kη>1/(1-x<kη>)

P(x)չ޺ȫʽɷ

P(x)1/(1-x<1η>) 
1/(1-x<ƽ>)
1/(1-x<>)
1/(1-x<4η>)

1/(1-x<kη>)


޺͵޻ͱˡ޻ǱɻʽɺP(x)ԡʾ

103(P<n>ɺP(x)ʽ)

P(x)1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)

һµĿǰΪֹĲ֣Ϊ˽ľʦ⣬ҲP<15>ҪһP<n>ȻΪץסP<n>ɺP(x)趨103103ʾѾ˻ʽɺP(x)
Ҫ˼һX

X
P(x)ݼչx<nη>ϵΪΣ

P(x)1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)

x<nη>ϵP<n>һP<n>102ĲʽP<15>1000

ֲһеͼ

ֲ      ɺP(x)
10-3 
ֲһP<n> ʽɺP(x)
         	    X
ͣ½⡣

򡡡򡡡

ͻưɡ׶֮̽˵
û
ţֻҪ֤102Ĳʽ÷ǵP<n>԰ɣ׶˵
šϡûұе㲻
˵ΪҲһP<n>ҲP<15>ܽ102׶ȵ˵
ף


1052׶ȵķ(ֲϽ)

Ҫ֤102ĲʽǷǵP<15>ɡ
׶ȱ˵վڰǰ棬ȡҵλá
ͬٵǉ˵ġǳֲP<n>ԼٶӵĺȵֲP<n>Ͻ硣
ʲôϽ磿ٵǉϷʡ
ϽǶn0P<n>M(n)ĺM(n)nʱP<n>ȻҲű󣬵ȴM(n)M(n)壬Ͻһ֡
ϷĽߣ˼𣿡ٵٽַͷϡ
ûϽȻڶϣﲢǶM(n)˵nĺ۲P<0>P<1>P<2>P<3>P<4>֮󣬾ͿԷǵ쳲F<1>F<2>F<3>F<4>F<5>

P<0>F<1>1
P<1>F<2>1
P<2>F<3>2
P<3>F<4>3
P<4>F<5>5

׶ֱȳ1123Ȼ5ͣ
ǺܿϧP<5>F<6>ΪP<5>7F<6>8ԡ

P<5>F<6>

ƲP<n>F<n1>ʽȻǡ

P<n>F<n1>

ʽʵ֤ȷʵҲϽM(n) F<n1>֤ѧɷ

򡡡򡡡

쳲ҳֲP<n>Ͻ
ֲΪP<n>112357쳲Fn0112358ʱn0ʽ

P<n>F<n1>

ʹѧɷ֤
ȣn0n1ʱP<n>F<n1>
֤k0

P<k>F<k1> P<k1>F<k2> P<k2>F<k3>

ɡ

֤£
ΪP<0>F<1>P<1>F<2>P<2>F<3>
ΪP<1>F<2>P<2>F<3>P<3>F<4>
ΪP<2>F<3>P<3>F<4>P<4>F<5>
ΪP<3>F<4>P<4>F<5>P<5>F<6>
Ҳ˵n0P<n>F<n1>ǶѧɷĽ˵Ϊͷðһʺŵĵٵǉ˵

ڼ֧k2Ԫķմ֧ʹõСӲֵĻ3Ρ

(1)ӲСΪٵΡ
(2)ӲСΪڵΡ
(3)ӲСΪϵΡ

֮²֧k2Ԫķ仯ɡ֧k1ԪķǱɡ֧kԪķ

(1)ӲСΪٵΣٳȥ1öʣµӲһɡ֧k1Ԫķ
(2)ӲСΪڵΣڳȥ1öʣµӲһɡ֧kԪķҴ֧֮СӲǢ١
(3)ӲСΪϵΣСӲΪ<m>Ȼ1ö<m>Ӳû

 ڣ٣٣
     ٹ(m-2)
ûȡ1öڣʣµӲҾͻɡ֧kԪķҴ֧֮СӲǢ١

˵ɽκΡ֧k2Ԫķɡ֧k1Ԫķǡ֧kԪķʱЩ֧ȫ죬Ҳ˵Щ֧˴˳ͻ
еѶk29ķֲزһεĻͻ±һõӲ˫߻ûĲ<>ʾ1Ĳԡʡԡ

P<9>			
9
81
72
711
63
621
6111
54
531
522
5211
51111
441
432
4311
4221
42111
4111
333
3321
33111
3222
32211

321111
3111
22221
222111
22111
2111
111
(1)P<8>Ĳ	
8<ɾ1>

71<ɾ1>

62<ɾ1>
611<ɾ1>

53<ɾ1>

521<ɾ1>
5111<ɾ1>
44<ɾ1>

431<ɾ1>
422<ɾ1>
4211<ɾ1>
411<ɾ1>

332<ɾ1>
3311<ɾ1>

3221<ɾ1>
32111<ɾ1>
311<ɾ1>
2222<ɾ1>
22211<ɾ1>
2211<ɾ1>
211<ɾ1>
11<ɾ1>	
(2)P<7>Ĳ


7<ɾ2>






52<ɾ2>



43<ɾ2>




322<ɾ2>









(3)P<7>Ĳ
<<ɾ2>11>



<6<ɾ2>1>


<5<ɾ2>11>













<33<ɾ2>1>









ֲڣԡ֧k2Ԫķĸᳬ֧k1Ԫķ롺֧kԪķĸܺ͡
ԴϵۣԵõk0ֲP<k2>P<k1>֮²ʽ

P<k2>P<k1>P<k>



P<k>F<k1> P<k1>F<k2>

ʱĽʽӾͻ

P<k2>F<k2>F<k1>

쳲ĶԵõұߵʽF<k3>ʽP<k2>F<k3>

˶k0

P<k>F<k1>ҡP<k1>F<k2>ԡP<k2>F<k3>


ѧɷõn0P<n>F<n1>

ţһ䣬ֲP<n>һֱ쳲F<n1>ѹסรûࡣǻû102F<k2>F<k1>F<k>쳲б

n   01234567 8 9 101112 13 14 15 16
P<n>01123581321345589144233377610987

ӱе֪F<16>987Իʽ

P<15>F<16>9871000

Ҳ˵

P<15>1000

102ĲʽãĽˡ
һP<n>ҲP<15>֤ˡ

102

P<15>1000

׶ؽķ


1053ٵٵķ

ǡǸλٵپ֡
õģٵǉ𣿡׶ָ
⡭ҲҪ102Ľ⡣ٵ˵
รǾͻְɡ˵꣬׶ȵݳ۱ʡ
ˡҵķܿͻҽ֧15Ԫķȫд˳ֱĻP<15>ֵ176ԡ

P<15>176<1000

102Ĳʽ
ٵ˵ţʼǱ̯

١15١13
١11ڡ2 ١9ڡ3
١7ڡ4١5ڡ5
١3ڡ6٣ڡ7
١12ۡ١10ڣ
١8ڡ2ۡ١6ڡ3
١4ڡ4ۡ١2ڡ5
ڡ6ۡ ١9ۡ2
١7ڣۡ2١5ڡ2ۡ2
١3ڡ3ۡ2 ٣ڡ4ۡ2
١6ۡ3١4ڣۡ3
١2ڡ2ۡ3 ڡ3ۡ3
١3ۡ4٣ڣۡ4
ۡ5 ١11
١9ڣܡ ١7ڡ2
١5ڡ3ܡ١3ڡ4
٣ڡ5ܡ ١8ۣ
١6ڣۣܡ ١4ڡ2ۣ
١2ڡ3ۣܡڡ4ۣ
١5ۡ2ܡ١3ڣۡ2
٣ڡ2ۡ2ܡ١2ۡ3
ڣۡ3ܡ ١7ܡ2
١5ڣܡ2١3ڡ2ܡ2
٣ڡ3ܡ2١4ۣܡ2
١2ڣۣܡ2ڡ2ۣܡ2
٣ۡ2ܡ2١3ܡ3
٣ڣܡ3 ۣܡ3
١10ݡ١8ڣ
١6ڡ2ݡ١4ڡ3
١2ڡ4ݡڡ5

١7ۣݡ ١5ڣۣ
١3ڡ2ۣݡ٣ڡ3ۣ
١4ۡ2ݡ١2ڣۡ2
ڡ2ۡ2ݡ٣ۡ3
١6ܣݡ ١4ڣܣ
١2ڡ2ܣݡڡ3ܣ
١3ۣܣݡ ٣ڣۣܣ
ۡ2ܣݡ ١2ܡ2
ڣܡ2ݡ ١5ݡ2
١3ڣݡ2٣ڡ2ݡ2
١2ۣݡ2ڣۣݡ2
٣ܣݡ2 ݡ3
١9ޡ ١7ڣ
١5ڡ2ޡ١3ڡ3
٣ڡ4ޡ ١6ۣ
١4ڣۣޡ ١2ڡ2ۣ
ڡ3ۣޡ ١3ۡ2
٣ڣۡ2ޡ ۡ3
١5ܣޡ ١3ڣܣ
٣ڡ2ܣޡ ١2ۣܣ
ڣۣܣޡ٣ܡ2
١4ݣޡ ١2ڣݣ
ڡ2ݣޡ ٣ۣݣ
ܣݣޡ١3ޡ2
٣ڣޡ2 ۣޡ2
١8ߡ ١6ڣ
١4ڡ2ߡ١2ڡ3
ڡ4ߡ ١5ۣ
١3ڣۣߡ ٣ڡ2ۣ
١2ۡ2ߡڣۡ2

١4ܣߡ ١2ڣܣ
ڡ2ܣߡ ٣ۣܣ
ܡ2ߡ ١3ݣ
٣ڣݣߡۣݣ
١2ޣߡ ڣޣ
٣ߡ2 ١7
١5ڣࡡ ١3ڡ2
٣ڡ3ࡡ ١4ۣ
١2ڣۣࡡ ڡ2ۣ
٣ۡ2ࡡ ١3ܣ
٣ڣܣࡡۣܣ
١2ݣࡡ ڣݣ
٣ޣࡡߣ
١6ᡡ ١4ڣ
١2ڡ2ᡡڡ3
١3ۣᡡ ٣ڣۣ
ۡ2ᡡ ١2ܣ
ڣܣᡡ٣ݣ
ޣᡡ١5
١3ڣ⡡ ٣ڡ2
١2ۣ⡡ ڣۣ
٣ܣ⡡ݣ
١4<11> ١2ڣ<11>
ڡ2<11> ٣ۣ<11>
ܣ<11>  ١3<12>
٣ڣ<12>ۣ<12>
١2<13> ڣ<13>
٣<14><15>
(ASCIIû10Ժţ<>档)
׶Ѹȷϵٵоٵ֧
ûдǵٵǉֵ׵ʤءЦ׶ٵٵͷ
ٺ٣ûˡٵٿĵ˵
ֻƿԡ


106

ΪصҵңͻȻú㡣
Լλ£ſϡ
ִһP<n>ǸܱʣĿѾرòʽˣһһʱɺԽȫûа
治ġ
߽ңŲ׶ȣĽŲ𽥽ӽ
ôˣ׶ȵ
ûлشҲû̧ͷ
ۡ
ڰĽУ׶Ծûκζ
һĬ
Ƶ̧ͷ
ı鲢ƽһУǴ飬ͻȻ䣬׶ȵָˡ

1123

쳲ƣǰѧ˴кķʽڵҲûӦ
׶Ȱֱڱ󣬽Ա˵
ٵǉܿɰء
ûлش
ܲһɰ
ҡûлش
ҵ˵¸˵ġ硻
ûнΪߴ·ˡ˵
š׶˵ţڵϵκεطԼ׳ʱ֮УѧǲϵѰŸĽм䣬նؽҲǳе£Ҫ˵Ѱû𣿵ȻǣȥѰҾ޷֪Ƿܹҵʵȥô֪Ƿܹˣƣʱ򡢻·ʱ򣬼ˣȻС
ҡװɺܶӣɺԽⲢûаһԳ
ô׶˵ôҷֵɺP(x)ɡ˵껰¶΢Ц
׶һָٶȰڳ쳲ơ

1123

ſӦԼơ

5

Ȼſֻλϵңůָҵӡ

ƣʱϢɣߴ·ʱ߰ɡΪһȫǵó̡

˵껰ǰսҡ
ǵ۾ײ
Ƭͫס
˵ӲϿ
ء
ʦֵĻһɡҲ˵롣
͡׶˵


107ѰҸϽ;

˼졣
ѧ׶ͻȻ˵
ҵ쳲õϽˣɣˣѵٵǉҲ


1071ɺ

׶ŷ۱վڽ̨ǰ档
ٵٺұгڽҵǰϿΡ֮ûκˡ
ҪֲP<n>Ͻ磬P<n>M(n)M(n)֮ǰѾ֤쳲ΪֲϽˣҪǸõϽ硣
νõϽָ쳲СϽ𣿡ٵپַʡ
ûn൱ʱˡ׶ȼ򵥵ػش
Ȼ۾˵ǵɺ

򡡡򡡡

ǵɺ˼ֲP<n>ɺP(x)ĴСϵ0x1ķΧ˼ĻP<n>x<nη>P(x)С

P<n>x<nη>P(x)

ΪɺĶаP<n>x<nη>ĹϵʽУұ߸Ϊ߻ұС

P<n>x<nη>P<0>x<0η>P<1>x<1η>P<n>x<nη>

ǻ֪ɺP(x)һʽûǻʽ(ʱ΢һ)ұ߿滻ʽ

P<n>x<nη>1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)

߳x<nη>

P<n>1/x<nη>1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)

ұߵʽӻP<n>ҪҲϽĺ򲹣޻ôҪnöҪ޻˼

P<n>1/x<nη>1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)1/(1-x<nη>)

ʽΪֹ㵥ұߵĻԾɺѴҪ΢һԡ
ģΪ鷳ɺͣҪôɺأ


1072һתǡɺ

ֻҪȡͺˣȡĻܽɺˡ˵
׶Ȼش

򡡡򡡡



P<n>1/x<nη>1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)1/(1-x<nη>)

ȡǡһתǡǴӼӡѰP<n>Ͻĵ·תѰS<eΪףP<n>>Ͻĵ·ٵǉûɣȻϸڵҲҪǲʧ˴

S<eΪףP<n>>ܩS<eΪף1/x<nη>1/(1-x<1η>)1/(1-x<ƽ>)1/(1-x<>)1/(1-x<nη>)> 

ȡ˶֮󣬻ͱɺͣõʽӡ

S<eΪףP<n>>ܩS<eΪף1/x<nη>>S<eΪף1/(1-x<1η>)>S<eΪף1/(1-x<ƽ>)>S<eΪף1/(1-x<>)>S<eΪף1/(1-x<nη>)>

ʽᷳæдɣ˼һ

S<eΪףP<n>>ܩS<eΪף1/x<nη>><k=1n,S<eΪף1/(1-x<kη>)>> 

ֳɶ·Ҳǡ·֮󻹻ҪǺá

S<eΪףP<n>><eΪףP<n>>
S<eΪף1/x<nη>><k=1n,S<eΪף1/(1-x<kη>)>>
	ߵɽ		    ߵɭ֡
ǰĻɽǰɭ֡

1073ߵɭ̩֡չʽ

ȿߵɭ֡ɡ

ߵɭ֡<k=1n,S<eΪף1/(1-x<kη>)>>

ߵɭnľҪ󹹳ɡߵɭ֡ġߵľҲǩS<eΪף1/(1-x<kη>)> Ͻ硣
ڵҪĺ

ߵľS<eΪף1/(1-x<kη>)>

˼ͬʱtx<kη>Ϊf(1)

f(t)S<eΪף1/(1-t)> 

Ҫоf(t)ôȽϺأ˵˵ٵǉôأ

򡡡򡡡

ףҡ𣿿һ̫SʲôԲ𡣡
в֪ĺf(t)㿴㣬ٵǉǡһӶ𣿡
̩չʽ
û׶˵f(t)̩չʽݼ

򡡡򡡡

f(t)̩չʽݼ
ڻõ΢Լϳɺ΢ֻ֣д
f(t)S<eΪף1/(1-t)>  ̩չʽΪݼ

ߵľS<eΪף1/(1-t)> 
t<1η>/1t<ƽ>/2t<>/20t1

ｫtԭxkͿԵõߵľݼչ

S<eΪף1/(1-x<kη>)>x<1kη>/1x<2kη>/2x<3kη>/20x<kη>1

ʽȡk123nĺͣҲǴӡߵľصߵɭ֡

ߵɭ֡<k=1n,>ߵľ
<k=1n,S<eΪף1/(1-x<kη>)>>

̩չʽ

<k=1n,x<1kη>/1x<2kη>/2x<3kη>/3>

ĺҲæʾ

<k=1n,<m=1,x<mkη>/m>>

͵˳

<m=1,<k=1n,x<mkη>/m>>

mĦԼԽ1/mᵽ档

<m=1,<k=1n,(1/m)x<mkη>>

ڲলչȷԼһ¡

<m=1,(1/m)(x<1mη>x<2mη>x<3mη>x<nmη>)>

м佻͵˳н͵˳ʱҪǳע⣬Ȳˡ
Ϣһ£ҪϽ磬Ҫҳȡߵɭ֡ʽӣｫ޺Ϊ޺ʽ޺ΪҪõȱȼ͹ʽǼǰɡ

ߵɭ֡<m=1,(1/m)(x<1mη>x<2mη>x<3mη>x<nmη>)>

ڲ޺޺͵Ĳʽ

ߵɭ֡<m=1,(1/m)(x<1mη>x<2mη>x<3mη>x<nmη>)>

0x<mη>1ȻʹõȱȼĹʽ

<m=1,(1/m)((x<mη>)/(1x<mη>))>

ٶͣ½Ų㲻ʽҲûϵΪҪϽ磬ֻҪҳʽӾͿԣҪע(x<mη>)/(1x<mη>)µķĸ1x<mη>ĸøСʽȡĻʽ
ǽʽӸ״롺΢һʽӡ״ʽЭ΢һϽ磬ÿЭʱͻֲȺš
۲졺ߵɭ֡ɡ

ߵɭ֡<m=1,(1/m)((x<mη>)/(1x<mη>))>

ĸʽֽ⡣

<m=1,(1/m)((x<mη>)/((1x)(1xx<ƽ>x<m1η>))>

ĸСx<m1η>Ժγɲʽ

<m=1,(1/m)((x<mη>)/((1x)(1xx<ƽ>x<m1η>))>
(ĸкһm)
Ϊx<m1η>mûʾ

<m=1,(1/m)((x<mη>)/((1x)mx<m1η>)> 

ʽ֮󣬵ٵǉม׶ȶԵٵ¶΢Ц
ף׶ѧ㣬ΪʲôһУ
Ǿ԰ɡ

ߵɭ֡<m=1,(1/m)((x<mη>)/((1x)mx<m1η>)>

ʽ

<m=1,(1/m<ƽ>)(x/(1x))>

ûܵƵʽƵ桭

(x/(1x))<m=1,1/m<ƽ>>


ɡ
⣡ʽǦ<ƽ>/6ٵٽе
û׶ʳָ

򡡡򡡡

ûڴĻмʹʦıĽɡ

<m=1,1/m<ƽ>><ƽ>/6

ټߡ

ߵɭ֡<k=1n,S<eΪף1/(1x<kη>)>>
(x/(1x))<m=1,1/m<ƽ>>
(x/(1x))(<ƽ>/6)

ڡߵɭ֡Ĺ۲͵
ˣΪԺķ㣬tx/(1x)ɣԡߵɭ֡ʾ

ߵɭ֡Ͻ

<k=1n,S<eΪף1/(1x<kη>)>>(<ƽ>/6)ttx/(1x)


1074ߵɽ𡻵

ǵóѾһ룬ص·ɣҪߵɽǰ
0x1ڿʼҪ֩S<eΪף1/x<nη>> 
͸ղһtx/(1x)0x1ʱʹ0txt/(1t)

ߵɽ𡻣S<eΪף1/x<nη>> 
n S<eΪף1/x>  ɩS<eΪףa<nη>>n S<eΪףa>õ
n S<eΪף(1t)/t> xtʾ
n S<eΪף11/t> 

ҪעS<eΪף11/t> u1/tȻ鵱u0ʱS<eΪף(1u)>Σ͵(Harmonic number)ʱʮƣȻͻġߵɽ𡻡

 [ͼƽֱϵu-y1/1+u(0,1)һֱߣȡֱһuӰɳΡεΪuӰu-yɵͼΪS<eΪף1+u>]


u0ʱбֵ߲Сڳεԡ

S<eΪף1u>u

Ϊu1/t

S<eΪף11/t>1/t

ɵõʽ

S<eΪף1/x<nη>>nS<eΪף11/t>n/t

ϾǶԡߵɽ𡻵Ĺ۲졣

ߵɽ𡻵Ͻ

S<eΪף1/x<nη>>n/t t0


1075;յ

ôһλص·ɣһһ㡣
۲졺ߵɭ֡롺ߵɽõĩS<eΪףP<n>>ͻʽӡ

S<eΪףP<n>>n/t(<ƽ>/6)tt0

ͿˣұߵʽΪg(t)g(t)t0ʱСֵСֵܽS<eΪףP<n>>ͷѹסˡ

g(t)n/t(<ƽ>/6)t
g(t)n/t<ƽ><ƽ>/6΢

ⷽʽg(t)0õt(<6n>/)t0ķΧڵõµ
t	0	  <6n>/   
	
g(t)          -      0         +
    
g(t)             С
Сֵ¡

g(<6n>/)(<6>/3)<n>

ñȽ׶ͼʾͻͼʽg(t)0ΪҪҳͼˮƽеλá

[ͼֻƽֱϵy=g(t)=(n/t)+(<ƽ>/6)t(<6n>/С3)<n>,(<6>/3)t]

ڵ߳Ĳˣڹĵֻn˵ĳͨͨΪK

S<eΪףP<n>>K<n>K<6>/3

ڡһתǡʱȡڷȡصתǾܿˡ

P<n>e<K<n>η>K<6>/3

ˣڸһˡ
Ȼọ́Ѿˡӭ

ֲP<n>Ͻ֮һ

P<n>e<K<n>η>K<6>/3

S<eΪףP<n>>Ͻ  еͼ

 S<eΪףP<n>> 

S<eΪף1/x<nη>><k=1n,S<eΪף1/(1-x<kη>)>>
	ߵɽ		       ߵɭ֡
 
ߵɽ𡻡·  ߵɭ֡
     
 n/t  n/t(<ƽ>/6)t  (<ƽ>/6)t
 
(<6>/3)<n> 


1076ٵٵĻع

Һ͵ٵһ׶ọ́ȻЩطȷϣҪƷ;֮󡭡׷ʽ֮޷ĸо
ҿٵ٣ڳı鲻һ
ٵ٣ѵй𣿡Сʡ
ôᣬȫйٵٿʵЦš׶ѧƵĹУкܶҲĵطûйΪҲкܶطö
ٵٵͷ֮˵
ܾúԹˣǺó;Ȼкܶطû꣬ѾץŵĸоˣҳҸеȤʹЩҲҾú

-޺ͱ޺ͣʽ
-γɼʽϽĹ۲Ϊϸ
-Ϊ˽Ͷȡ
-ʹ͹ʽ
-ƿʱ̩չʽ
-鷳ʱı任
-ʦı
-ΪСֵ΢ֵ

֮ҪԼĥȻս⣬ŻԾʵСǽȻͺãҪĸжȥӡһתǡ·Ȼǡߵɭ֡롺ߵɽ𡻡ԼȥѰңѧϰ֪ʶ׶ѧ㣬лл㣬Ȼһû취ÿúõãʹ֮ǰҪȡЩ֪ʶɣһ͵ġ
ٵٽȭͷ


108ټ

ڻؼҵ·ϼѧǸղŵϽ磬Ҫõ쳲иõϽĻnҪ󣿻ҪP<n>ۣٵ˷ܵһֱ⣬Ȼһشʱ׶ҲĽ顭ǲϻ
󡭡ͨһĵ·һĳվ
ƽ׶ȶѸ뿪ٵȴҪһߡ
ףٵ٣Ϊʲôߣ
ٺ١Ҫ׶ѧһȥꡣ
ԭˡܺ
ˣ׶˵
ѧѧɡ
ٵٴ˵󣬾ͺ׶һ𲢼ǰߡ

ȥ
һˡ

ףոջһ̸ֱȴͻȻֻʣһˡܾúеį
ͬһѧһԵĵ·ôǵʱռȻޣս٣ҵؿڴһʹ
Զĵٵ׶ȶ˵ЩʲôͻȻ䣬תҡ

ô

ٵٸ߾֣ҵҡڡ
׶Ǿؾ֡
ȻϽ࣬ͬʱڶָ
һһٵٵ

쳲ƣ˻ͬʱ
¶Ц
ûʱȷʵޣҲս٣ΪˣŻȥѧǰǵΪѧԡ
ΪѧǳԽʱġ

߾˫֣ӦλѧŮ

׶ȡ
ٵ١
һѧɣ

˵µĽ
ǿֿ˵ǴӴԺҸֵӡ
ǶǶԣµĿʼ
ǵðգ
Ȿķ⡣
CS˹սۡ(롡־룯)



β


졣
ʦ
һλŮܽ˽ʦ칫ҡ
ʦ㿴㿴ѧ±ɢˡ
ǰΪѧˡԱأ
СдӲģP<15>176Ա1000С֤ϣ˰ɣʦ
Ů̯ʼǱ
ȷ𰸡ԭˣȫдȥ˰
û취ͷԽĻֻܶˣ176Ҳ1000̫Զ˰ɣˣʦбʾֲһP<n>ʽ𣿡
ţаɡ

X(ʾֲһP<n>ʽ)

P<n>(1/(С<2>))<k=1,A<k>(n)<k>(d/(dn))(sinh(<n-1/24>/k)/<n-1/24>)>
C 
ʦŵʽʲô
ܾ˰ɣ÷һ귢ʽӡ
มһ¡Ak(n)ʲôûж尡
ţע⵽ˣϸʽȻʦû취һ仰ϸ˵A<k>(n)Ǵĳ޺͵óģ޺͵24η1ϸĻҪսˡ
Ҫ̶ֳ𡭡
֮ķֲл಻˼ġม

ʦȱѧƬʦŮ𣿵ص㡭ŷ𣿡
˺ˣҪԿ˵š
ףһǲһŮƬձأ
ιߡ
ʦܻӭأŮǺǵЦ
ʦҪѣǴӸʱʼһѧеͬ顣
รԭʦҲиʱ
⵱Ȼˣȥȥ
µĿƬҾͻȥ

ҵݳƬŮֽӹ
ףʦſƬ
ţĿƬһġ
áǾʹˣ
Ů΢ЦŰڶָĴΡ
ҽſӦ߳ʦ칫ҡ

찡
ʦ칫Ҵӣһʱ

޵չ
÷ḻʵĲҵ
ҪȴߵŬ
[25]






ǽǺơ
Ҷѧġ㽡кŮֵĳЩطеơ
ѵѧȴҲ𰸣һͷûУ޷Ȿ֮ɵزĳá
֪飬ϲ޷֪ҷǳԺǸӰ
Ȿûн鴫أ

ڶдԭͣվϣµĶĵ֧ңûЩԣͲ 
֡ѧŮⱾˣһҵĸл֮⡣
ʹLATEX2Euler Font(AMS Euler)Ű棬֮Euler FontͣдһֵֺѧҺնշ 
дƣרŸõ͡
Űϸл´ġLATEX2š˺ܴó
һֵͼʹ˴һ(tDB)ĳѧץɥޥmathJoyJ÷룬ץ=printӡˢƷӡˢޥ=macro桭Ǹ֣ӡƶɣǷǳл(ఴָԭ)
лĶԭ壬豦ĸλ

ľľԭ¡ƽֲʸʮ(о)ǰԭ 
ӢլϲʸҰɽڽʷС㡣

ҲлеĶߡҵվǣԼΪĻ̽ѡ
лڱ֮ǰޱ֧ҵҰϲܱ༭Լ󻮱ʱЭеС㡣
ҲлԼӣرǸԭ彨ĳС
Ȿ׸ҵʦ-
ǳлĶ飬ϣлټ

Ǻ
2007꣬300Ĵ
http://www.hyuki.com/girl/



οƼ鼮


οƼ鼮Ȼ·ʽֻ࣬Ǹţע⡣

-һ
-
-ѧ
-оרҵ
-ҳ


һ

[1]GPolya룬ˤȤʽ磬ISBN:46210336891954ꡣ(ǣ⡻룯ĻISBN:978986417724920060626ա)
ѧΪν⣬ʷԵ˵ѧϰ˱ضһ顣
[2]gۣ-ѧˤʤ뱾JoyJ-ѧֵ飩̸ִ飬ISBN:40614984012006ꡣ
Сѧѧѧѧġƿ緽ʽʽֵȵȣѧϰѧ˳ֵ 

[3]ǺƣץѧJoyJѧեȥХ󥯥ꥨƥ֣JoyJSoft Bank CreativeISBN:47973297342005ꡣ
ѧϰʱóġѧ˼ʽѧϰ飬Ҳ˵߼ѧɷϡ֤ȵȡ 
http://www.hyuki.com/math/
[4]Doug1as RHofstadterҰѺ룬`ƥ룬å`Хåϡ뤤ϲ˼λJoyJ¶Ѧ͹-Ĳ˼֮磬ISBN:48269002521985ꡣ(ԭGodelEscherBachAn Eternal Golden Braid)
Ը¶Ѧ͹Ϊ⣬߼ìܡݹ顢֪ʶ˹ܵȵĶ׶ӢӢףǲο20ġSheppardס⣬г桺20桹(2005)
[5]Douglas RHofstadter۵룬᥿ޥå`ࡪѧܿΥ`ѥ롻JoyJMeta Magic Game:ѧδ֮աMeta-ǰ׺ʾУ֮⣩磬ISBN:48269 004301990ꡣ(ԭMetamagical Themas)
ڿѧ־еļأħĽⷨ⣬ĻⷶΧ൱ع㡣⣬г桺20桻(2005)
[6]Marcus du Sautoy룬֡³磬ISBN:41059004982005ꡣ(ԭThe Music of the Primes)
ѧҷȡԡ֡͵ĽǶдرǺĹӡ̡
[7]EAFellmannɽ֮룬`ĤҵJoyJ-Ṧΰҵץ󤬩`ե`ISBN:44317092822002ꡣ(ԭLeonhad Euler)
ĴǡڸλԾԼΧλģ[8]δѧڱίԱ࣬17ഺ 
2006׺УΥå`JoyJ17ഺ2006-׺ϢС塱17ָٽ䡣ٽ䣬ձͳʫʽ֮һйɣһ5ڣڶ7ڣ5ڣNHK棬ISBN:41401614262006ꡣ
ԡδѧȫٽ͡Ϊٽ伯57517Ҳ




[9]д̣եܥʥåС桻JoyJ쳲еС棩ձ磬ISBN:45357828142002ꡣ
дӳݵרŵĶԣ쳲еһ
[10]ңУѧJoyJ=£棬ISBN:97843200176892004ꡣ
Ѽмһִѧѧ飬վҲԶݡhttp//www.h6.dion.ne.jp/~hsbook_a/
[11]Ҳ壬ƺǧۣޥ`֩`ϤJoyJMaster of ϤϤʲ÷룬ᾫɣ棬ISBN:48874202851999ꡣ
ϵĸòο飬гֿC<n>Ȥ⣬ڡѧŮУڶӦ·ͨĿһ󷨾ǲο顣
[12]־غƶѧäƤ1 ޤߡJoyJѧĹ1֮Ҳ꣬ISBN:40000791151994ꡣ
˵С޺ݼ飬ֻĶʽ͸ʦѧĶԻѧϰ֪ʶȻһܱ飬ȴ롣
[13]´ȣJavaˤ륢르ꥺµ䡻JoyJJavaе㷨ɣ磬ISBN:47741172932003ꡣ
㷨ΪJavaĴǵ䣬Ϊֲʱĵƹʽǲο顣
[14]William Dunhamڴأɽˣ١Ҳ룬`šJoyJţץ󤬩`ե`ISBN:44317107952004ꡣ
ѡڸѧо飬ص˼뷨Ϸ绯ķʽ֡ڡѧŮУ¡¡ۡкܶطο顣
[]5]СߣʤĤȤ륪`ȥեީ`JoyJѶԻ¼̸磬ISBN:406154537X2003ꡣ
ѼȤѧ⣬Ҳн˵֤֮⣬(2)ֵ֮ķ
[16]George EAndrewsKimmo Erikssonĵ롺ηָѧ鷿ISBN:48269310342006ꡣ
ԷֲΪ顣(дGeorge WAndrewsGeorge EAndrewsӡˢ)ΪֲеȨӷֲŵ鱨ϸĽ˵Ȿھĩ¼н޻ͳڡѧŮʮУ׶쳲ֲ֤Ͻʹñp2931֤
[17]ڴأ``ޥ󡢥ޥ̥󡻣JoyJūҲ꣬ISBN:40000746602006ꡣ
ūΪ⣬಻˼
[18]ʽ`JoyJѧܿĿ⣬ISBN:44800867572001ꡣ
Ϊʽ 1ۻѧ飬ĿⱾ˶ʽ൱
[19]ʽѧȫλѧJoyJ-ѧʦͨѧᣬISBN:44860148552000ꡣ
ѧΪģӻʼ䷳ضʵʺϻѧϰݸѹԵȤζѧŮڶУʽʽĻǲο顣


ѧ

[20]Ƚһʤ֤ѧ-С˷饦`֥åȤޤǡJoyJ׶Ӧѧ-С˷ʼ`֥åȶաòˣ棬ISBN:43200173822003ꡣ
ӸѧѡݽҪõѧѭ򽥽ʦԻķʽ⡣ѧŮйϣֵĲּΪο顣
[21]Ronald LGrahamDonald EKnuthOren PatashnikϣͨΣҰҲʯx룬ԥ`ѧJoyJѧ棬ISBN:43200266831993ꡣ(Ŭ˹ϣˣ룯ֳ硻ѧ)
͵ɢѧ鼮Dӡݽ׳ˡеۻʹɺһķȽԲο飬ڡѧŮڳֵģڱи꾡Ľ˵
[22]Donald EKnuthϵ룬The Art of Computer Programming Volume 1ձ桻JoyJ̵1ʽ祢`ISBN:475614411X2004ꡣ
Ϊ㷨ʷԽ̿飬128н˷ֱչʽЧɺ232н˴΢ʽķеʽ͵ļȵ롺ѧŮ൱ԵĻ⡣
[23]Donald EKnuthTheArt of Computer ProgrammingVolume4Fascicle 3Generating All Combinations And PartitionsJoyJ̵4£AddisonWesleyISBN:02018539492005ꡣ
ֲصĸ㷨ͳѧԽ鼮Ҳο7214 Generating all partitionsڣرThe number of partitionsһСڡ
[24]Jiri MatousekJaroslav NesetrilҲбغ룬ɢѧؤд()ץ󤬩`ե`ISBN:44317089792002ꡣ
ɢѧȵȤζ⡣ڡѧŮʮУ׶õϽǲο鶨1072֤(p129)
[25]Leonhard Euler룬`޽磬ISBN:48752520212001ꡣ
-Լ׫дй鼮ᵽʹ޺޻ȤıǷʽeҲеǳþʽڷܼģ˼άԽʱسǰ


оרҵ

[26]Richard PStanleyEnumerative CombinatoricsJoyJѧVolume 1ISBN:05216635121997ꡣ
ѧĽ̿顣
 [27]Richard PStanleyEnumurative CombinatoricsVolumc 2ISBN:05217898771999ꡣ
ѧĽ̿顣رʺϰÿ(Catalania)࿨Ļʵ(pp219229)
[28]Ԫ`ޥΤ`JoyJ꣬ISBN:42541173102005ꡣ
飬Ҳο14ͷİķ֤ĵͼɢԼ޻ʾ()Ե֤
[29]ڴأ`о꡹JoyJоձ磬ISBN:453578334462002ꡣ
ܹڦƸ鼮ӦǼѧ⣬ʵʶȴԣˬе
[30]Hans RademacherA Convergent Series for the Partition Function p(n)ProcLondon MathSoc43pp2412541937ꡣJoyJɢĻУ
չʾֲһP<n>ġ


ҳ

[31]http://www.research.att.com/~njas/sequences/Neil JASloaneThe OnLine Encyclopedia of Integer SequencesJoyJе߰ٿȫ飩
еİٿȫ飬뼸֮󣬻ʾС
[32]http://scienceworld.wolfram.com/biography/Euler.html
򵥽ҳ׶ȴӴҳų̀ʣԭԭ֮¡
He calculated just as men breatheas eaglessustain themselves in the airJoyJļһ㣬ͬ𰸾ڿ֮У(by FrancoisArago) 
Read Eulerread Eulerhe is our master in everythingJoyJĶɣĶɣз䣩(by Pierre laplace)

[33]http://www.gakushuin.ac.jp/~881791/mathbook/ѧѧӘSिˡJoyJѧ-Ϊ˿ֵѧϰ
ѧΪƶѧ̿飬PDFʽҳϣοʽ˵ķ
[34]http://mathworld.wolfram.com/CatalanNumher.htmlEric WWeisstein et al.Catalan NumberFrom Math WorldA Wolfram Web Resource
ڿҳйصƹʽʽϵĹϵԼʵ
[35]http://mathworld.wolfram.com/Convolution.htmlEric WWeissteinConvolutionFrom Math WorldA Wolfram Web Resource
ҳ
[36]http://www.hyuki.com/girl/ǺƣѧŮ
ѧŮĶվСѧŮ鱨

Ϊϲѧϰ
صȴʦȴϿΣ
ֻҪѰҡֻҪĶ
ѧϰԶ̡
ѧŮ[36]



(֮ΪԭDOCδ)


šӢ 

a priori216
Convolution156
emath307
Eulerեȡ307
Harmonic number293
(omega)527
(pi)7262
(product۳)29
rationale19
(sumۼ)29
(1)228
(2)228


(b)

⡡233292
չ65
ǡ27


(p)

ƽ֡123132196
ƽ90


(m)

׶ȡ5
⡡165
ݼ209
ݼչ214


(f)

182
֤198
ֲ248
ֲ248
쳲ơ255
쳲С664274
ɢ163180
ǡ46
ƽ桡43


(d)

229
塡1718113
ʽ32
ٵ١15
λԲ45
ȱȼ61
ȱе͹ʽ2960182
ȷֵ㡡46
ȲС49
ƹʽ64
ݽ׳ˡ108128187188
γɵĹ廯123210
87130222
215


(t)

ͶӰ47
̩չʽ219
ͼ184228
184


(l)

÷304
ǹʽ41
ĻƽĲ57239
183
54
120


(g)

廯5391118123125231

(k)

Cn156
ж39262


(h)

ʽ3181209
նշ307


(j)

ֵ22
Բʽ81
ⷽʽ37
ޡ101162
18
Խʯ1988125165


(q)

ߡ101


(x)

35


(zh)

ߡ211
תӰӡ47
ۻ156
ӷֽ⡡30
ӷֽΨһԡ19197
17
ͬ36263


(ch)

֡104
Ӧ104
ݡ26


(sh)

Ͻ硡272
ɺ62132
ֻ195
162
ʹɺеһ64
ɾں()163210
ѧɷ274
ûȷ𡻡9
ʵ88117125


(r)

⡡86


(z)

 nCk127


(c)

Ȼĵطʼ81165


(s)

ʽ须38
ƽ뼸ƽĹϵ87


(y)

201207243305307308
ʽ34
ӡ34
㡡74121131


(u/yu)

ޡ10210
޴160
΢֡101213
΢D102


(/yu)

Լ165
Բʡ7262


(ou)

ŷá199


(er)

ʽ130262



