1、Weierstrass 逼uni8FD1uni5B9Auni7406“暫時uni7684uni6C92落, 不是uni6C38uni9060uni7684埋uni6C92。”uni8B39以uni6B64文紀念 Karl Weierstrass (1815-1897) 逝世uni4E00uni767E週年uni6797uni7426uni711C1、uni8FD1代分析之父Karl Weier-strass :“英uni96C4出uni5C11年”uni9019似乎是數uni5B78uni53F2uni751A至是整個科uni5B78uni53F2uni7684普遍uni5B9A律, 從牛un
2、i9813、Euler、 La-grange、Laplace、uni9AD8斯等uni9019些uni5927數uni5B78uni5BB6,uni5728很年輕uni7684時候便已展uni9732其才華。 uni5C0D於uni9019些uni5929才, 我們uni9664了uni8B9Auni5606之uni5916, 坦uni767D而言他們uni96E2我們uni9019些凡uni592B俗uni5B50是uni592A遙uni9060了。 uni5982uni679C要像uni9019些人uni7684成uni5C31才算偉uni5927uni7684話, uni90A3uni9
3、EBC我們uni6C92有幾個人uni7684uni751Funi547D是有價值有意義, 更遑論我們所作uni4E00uni9EDEuni9EDEuni7684數uni5B78工作。uni5728數uni5B78uni53F2上出uni73FE Karl Weierstrass, uni8B93我們感uni5230uni975E常uni7684欣慰, uni56E0uni70BAuni4E00個人uni7684成uni529F不見得是牛uni9813或uni9AD8斯uni7684方式, 也uni53EF以是 Weier-strass uni7684方式。Karl Weierstrass 1
4、815年uni751F於德uni570Buni7684Ostenfelde , uni5927uni5B78時代uni5B78uni7684是uni6CD5律與uni8CA1經,後來改uni8B80數uni5B78但uni6C92有uni62FFuni5230uni535A士uni5B78位。 uni5728uni56DB十歲以uni524Duni90FDuni9084uni5728中uni5B78教書, 後來uni7531於所uni767C表uni7684文章uni95DC於 Abel uni51FD數uni7406論, 才uni53D7uni5230uni5927uni5B78uni768
5、4uni6CE8uni76EE, 他uni5728普魯士uni9AD8等uni5B78uni6821任教uni591A年之後,才uni57281856年成uni70BA柏uni6797uni5927uni5B78數uni5B78uni7CFB教uni6388, uni4E00uni76F4uni5230uni53BB世。uni6B63uni5982uni53E4希臘uni54F2uni5B78uni5BB6入uni9580uni7684教訓 “認uni8B58你自uni5DF1”, uni4E00個人uni53EA有uni5728uni6B63uni78BA認uni8B58自uni5DF1之後
6、, 才有uni53EF能uni7372得成uni529F。我uni76F8信 Weierstrass是認uni8B58他自uni5DF1, 不像uni9AD8斯、 Cauchy、 Rie-mann 等數uni5B78uni5BB6uni90A3uni9EBCuni5929才橫uni6EA2有快uni901Funi7684uni76F4覺,Weierstrass uni5C0D數uni5B78uni7684uni7814究uni6CE8uni91CD方uni6CD5處事uni56B4uni8B39。 Weierstrass uni7684uni540D聲主要是基於他uni6975uni7AEF細
7、心uni7684推論方uni6CD5, 人們稱之uni70BA “Weierstrassuni7684uni56B4uni683C”。Weierstrass 早uni671Funi7684工作是uni95DC於超uni6A62uni5713積分 (hyperelliptic integrals)、 Abel uni51FD數1112 數uni5B78傳uni64AD 22uni53773uni671F uni6C1187年9月與代數微分方程。 但他最廣uni70BA人uni77E5uni7684uni5247是uni5229uni7528冪級數 (power series) 來建構uni890
8、7uni8B8Auni51FD數uni7406論, 而其中他最有興趣uni7684uni90E8份uni5247是整uni51FD數(entire function) 與uni51FD數uni7684uni7121uni7AAE乘積。uni5747勻收歛 (uniform convergence) 是他uni767Cuni73FEuni7684、平uni9762上uni7684解析uni51FD數稱uni70BA整uni51FD數uni5C31是他uni53D6uni540Duni7684、線性代數uni7684行列式 (determinant) 是他uni9996uni6B21以uni77E
9、9uni9663之元uni7D20所形成uni7684齊uni6B21uni591Auni9805式來uni5B9A義。uni53E6uni5916他uni5C0D於uni96D9線性型式 (bilinear form) 與uni4E8Cuni6B21型 (quadratic form) 也有著不uni53EFuni62B9uni6EC5uni7684uni8CA2獻。uni5C0Duni4E8C十世紀uni7684數uni5B78而言,他uni548C他uni7684uni5B78uni6D3E所uni5F15介uni7684“分析算術uni5316”(arithmetizationof a
10、nalysis) uni7121疑是最uni91CD要uni7684。他從不uni8AF1言曾經uni5728中uni5B78教uni904E書。 uni7531於uni9019uni6A23uni7684經uni9A57, uni5C31算是uni5728uni5927uni5B78教書也是uni4E00位uni597D老師, 他uni7684uni8B1B義uni90FD是經uni904E細心uni6E96備, 所以享有uni6975uni9AD8uni7684聲uni8B7D, 而他uni7684觀念主要是uni901Auni904Euni9019些uni8B1B義而成uni70BA數u
11、ni5B78uni5BB6共有uni7684uni8CA1uni5BCC。分析中 - uni7684觀念是他所提出uni7684。 uni7531於他作uni5B78uni554Funi7684才華uni548C教書uni7684uni672C事uni5438uni5F15了很uni591Auni5B78uni751Funi5230柏uni6797來, uni9019些弟uni5B50中最出uni540Duni7684有 H. A. Schwarz(1843-1921)、 Sofya Kovalevskaya (1850-1891)、G. Cantor (1845-1918)、 MittagL
12、eer(1846-1927),uni9084有 D. Hilbert(1862-1943) uni672C世紀最偉uni5927uni7684數uni5B78uni5BB6之uni4E00。 其中最令人稱uni9053uni7684是, uni5728他uni90A3個時代uni7834例收 SofyaKovalevskaya 或 (Sophie Kowalevski) uni70BAuni5B78uni751F(uni7531於柏uni6797uni5927uni5B78uni62D2收uni5973性, Sofya uni7121uni6CD5入uni5B78且uni9032入課uni58
13、02聽uni8B1B), 他自uni5DF1uni6BCF星uni671F日下uni5348uni64A5出uni4E00uni6BB5時uni9593uni5728uni5BB6uni88E1uni7279uni5225指uni5C0E她uni5B78習數uni5B78, 最後並推uni85A6她uni7372得德uni570Buni54E5uni5EAD根uni5927uni5B78uni7684uni535A士uni5B78位(Sofya Kovalevskaya 是uni54E5uni5EAD根uni5927uni5B78有uni53F2以來第uni4E8C個uni5973uni535
14、A士)。 Weierstrassuni5927uni6982是數uni5B78uni53F2上uni9AD8等數uni5B78最偉uni5927uni7684教師uni5427!Weierstrass uni5C0D於uni9023續uni51FD數uni7684uni7814究有絕uni5C0D性uni7684影uni97FF。 其中之uni4E00uni5C31是他uni9020了第uni4E00個處處uni9023續處處不uni53EF微分uni7684uni51FD數, uni7531於uni9019個結uni679C使得uni524Duni9762uni8AF8uni591Auni5
15、C0Duni9023續uni51FD數uni7684uni8AA4解得以uni91D0uni6E05。uni53E6uni5916uni5247是uni9019uni7BC7文章所要uni63A2討uni7684 Weier-strass 逼uni8FD1uni5B9Auni7406: 任意uni5B9A義uni5728uni5BE6數軸上之uni9589uni5340uni9593uni7684uni9023續uni51FD數uni90FDuni90FDuni53EFuni85C9uni7531絕uni5C0D且uni5747勻收歛uni7684uni591Auni9805式表uni793A
16、之。 uni9019個uni5B9Auni7406Weierstrass uni5728中uni5B78教書時uni5C31uni767Cuni73FE。 uni53EF見環境並不是作uni7814究uni7684uni552Funi4E00uni6C7Auni5B9Auni56E0uni7D20, 個人uni7684uni6BC5uni529B與uni57F7著才是更uni91CD要uni7684uni52D5uni529B。回uni9867uni4E00下; uni5728uni9AD8等微積分uni95DC於uni4E00致收歛或uni5747勻收歛 (uniformly converg
17、ent) 有uni4E00個很uni91CD要uni7684uni5B9Auni7406: 若有uni4E00uni5E8F列uni5747勻收歛uni7684uni9023續uni51FD數,uni5247其uni6975uni9650必uni5B9A是uni9023續uni51FD數。uni63DBuni53E5話說uni9023續uni51FD數uni5728uni5747勻收歛下有uni5B8C備性。 uni901A常我們最有把uni63E1也最有興趣uni7684是uni591Auni9805式uni51FD數, uni56E0uni6B64上uni9762uni9019uni6BB
18、5uni6558述uni53EFuni63DBuni70BA: 若有uni4E00uni5E8F列uni5747勻收歛uni7684uni591Auni9805式uni51FD數,uni5247其uni6975uni9650uni4E00uni5B9A是uni9023續uni51FD數。uni7531uni6B64自uni7136而uni7136uni53EF以uni554F其逆uni6558述是uni5426成立uni5462? uni63DBuni53E5話說, uni5728uni5340uni9593 a,b uni7684任意uni9023續uni51FD數是uni5426uni53
19、EF以uni53D6uni591Auni9805式uni51FD數來逼uni8FD1uni5B83uni5462? uni9019uni5C31是著uni540Duni7684Weierstrass 逼uni8FD1uni5B9Auni7406 (Weierstrass ap-proximation theorem)。Weierstrass 逼uni8FD1uni5B9Auni7406uni4E00: 任意uni5B9A義uni5728有uni754Cuni9589uni5340uni9593 a,b uni7684uni9023續uni51FD數 f, 總是uni53EF以uni7528un
20、i591Auni9805式Pn 來逼uni8FD1而且兩者之uni8AA4uni5DEEEn(f)= maxaxb|f(x)Pn(x)|=|fPn|uni7576n 時趨uni8FD1於uni96F6。Weierstrass 逼uni8FD1uni5B9Auni7406 13uni9019uni5C31是Weierstrassuni5728上個世紀末uni8B49明uni7684uni5B9Auni7406。 其uni91CD要性uni5728於保uni8B49任意uni7684uni9023續uni51FD數uni90FDuni53EF以uni7528uni591Auni9805式來逼uni
21、8FD1, 而且uni53EF以uni9054uni5230任意要uni6C42uni7684精度。 uni9019個結uni679Cuni5C0Duni51FD數uni7684uni8FD1似uni7406論有很uni5927uni7684幫uni52A9, uni540C時也說明了uni70BA何uni591Auni9805式uni53EF以形成uni901Auni7528uni7684uni8FD1似uni51FD數之uni7406uni7531。uni9664了代數uni591Auni9805式之uni5916, uni53E6uni4E00uni8FD1似uni51FD數uni768
22、4形式是uni4E09角uni51FD數, 或者更uni6B63uni78BAuni5730說; uni7528“Fourier 級數” 來逼uni8FD1uni9023續uni51FD數, uni9019也是Weierstrass uni7684uni8CA2獻。Weierstrass 逼uni8FD1uni5B9Auni7406uni4E8C: 任意uni5B9A義uni5728, 且週uni671F等於 2 uni7684週uni671Funi9023續uni51FD數 funi6C38uni9060uni53EF以找uni5230uni4E09角uni591Auni9805式Sn =a
23、0 + (a1 cosx+b1 sinx) +(an cosnx+bn sinnx)使得兩者uni7684uni8AA4uni5DEE|f Sn| 是任意uni5C0F。uni9019個uni5B9Auni7406uni53EF視uni70BAuni5B9Auni7406uni4E00uni7684uni7CFB (corol-lary), uni539Funi56E0是我們uni53EF以uni5229uni7528uni4E09角uni8B8Auni63DBuni5C07 Snuni8F49uni63DBuni70BA cosx 與 sinx uni7684uni591Auni9805式。
24、 Weier-strass逼uni8FD1uni5B9Auni7406uni7684uni76F4接應uni7528uni9996先uni5C31是uni51FD數uni7684uni8FD1似uni7406論 (approximation theory)。 uni9019個uni5B9Auni7406uni5C0Duni51FD數論而言是個不uni53EF或uni7F3Auni7684工具。 uni5728uni8907數域也有uni985E似uni7684uni5B9Auni7406。 1937年 M. H.Stone uni70BA了解uni6C7Auni62D3樸空uni9593中之u
25、ni9023續uni5BE6值uni51FD數中某些代數性uni8CEA之uni554Funi984C, uni767Cuni73FE Weierstrass逼uni8FD1uni5B9Auni7406之uni4E00般uni5316, 今uni5929我們稱uni70BA StoneWeierstrassuni5B9Auni7406, 應uni7528uni6B64uni5B9Auni7406使得uni4E00些uni91CD要uni7684uni53E4典uni5B9Auni7406之uni8B49明uni8B8A得簡uni55AE。 uni5C0D於 StoneWeierstrass u
26、ni5B9Auni7406有興趣uni7684uni8B80者uni53EFuni53C3uni95B1uni5BE6uni8B8Auni51FD數論與uni6CDBuni51FD分析uni7684書, 或者uni76F4接uni770B Stoneuni7684文章 (請uni53C3考 2、7) 。uni9019文章是作者uni63A2uni7D22數uni5B78中之某uni4E00主uni984C時經uni9A57uni7684記uni9304與uni8B80書心得, uni9019是uni4E00uni6B21令人興奮之uni65C5而我希uni671B能把uni5B83uni5BE
27、B得明簡, uni540C時也希uni671B透uni904E Weierstrass uni7684簡介, 能再uni6B21uni6FC0uni52F5我們uni5B78習數uni5B78uni7684心志。數uni5B78是打uni958B科uni5B78uni5927uni9580uni7684uni9470uni5319, , 輕視數uni5B78必uni5C07uni9020成uni5C0Duni4E00切uni77E5uni8B58uni7684uni640Duni5BB3,uni56E0uni70BA輕視數uni5B78uni7684人不uni53EF能uni638Cuni63
28、E1其uni5B83科uni5B78uni548Cuni7406解uni5B87uni5B99萬uni7269。 Roger Bacon 2、 Bernstein uni7684方uni6CD5 :Bernstein(1912) uni7684uni8B49明方uni6CD5是建構性uni7684 (constructive): uni76F4接uni9020uni4E00個uni591Auni9805式來逼uni8FD1uni9023續uni51FD數, uni9019個uni591Auni9805式我們uni73FEuni5728稱uni70BABernstein uni591Auni98
29、05式。2.1 Bernsteinuni591Auni9805式2.1 uni5B9A義: 任意uni9023續uni51FD數 f C0,1,其uni5C0D應uni7684 Berstein uni591Auni9805式uni5982下:Bn(x;f) nsummationdisplayk=0parenleftBiggnkparenrightBiggf(kn)xk(1x)nk(2.1)uni9019個uni591Auni9805式uni7576uni7136不會憑空想像uni5C31出uni73FE(uni9748感?) 數uni5B78最uni91CD要uni7684是思維uni768
30、4方uni6CD5, uni5982uni679C你要等待“uni9748感”, uni90A3uni9EBC你uni5C31uni6C38uni9060uni64AD不了種也uni7121uni6CD5收uni5272, uni7121uni7DE3享uni53D7uni77E5uni8B58uni7684uni6A02趣,明uni767Duni9019uni7576中uni7684奧妙。 Bernstein uni591Auni9805式uni7684uni8D77uni6E90uni5C31是uni4E8Cuni9805式uni5B9Auni7406:(x+y)n =nsummation
31、displayk=0parenleftBiggnkparenrightBiggxkynk (2.2)14 數uni5B78傳uni64AD 22uni53773uni671F uni6C1187年9月上式uni5C0Dx微分後乘 xn 得x(x+y)n1 =nsummationdisplayk=0parenleftBiggnkparenrightBiggknxkynk (2.3)再uni91CDuni8907uni524Duni9762 (2.3) uni7684計算uni904E程uni53EF得x2(x+y)n2 + xyn (x+y)n2=nsummationdisplayk=0pare
32、nleftBiggnkparenrightBigg(kn)2xkynk (2.4)令y = 1x, uni5247uni7531 (2.2)-(2.4) uni53EF分uni5225得uni9023續uni51FD數 1、x、x2 uni7684 Bernstein uni591Auni9805式Bn(x;1) =nsummationdisplayk=0parenleftBiggnkparenrightBiggxk(1x)nk = 1Bn(x;x) =nsummationdisplayk=0parenleftBiggnkparenrightBigg(kn)xk(1x)nk=xBn(x;x2)
33、 =nsummationdisplayk=0parenleftBiggnkparenrightBigg(kn)2xk(1x)nk=x2 + 1nx(1x) (2.5)uni5C0D於第uni4E09uni9805作個估計:vextendsinglevextendsinglevextendsinglex2 Bn(x;x2)vextendsinglevextendsinglevextendsingle = x(1x)n 14n n(2.6)uni56E0uni6B64uni7576n很uni5927時Bn(x;x2) 與x2 是uni975E常接uni8FD1, uni63DBuni53E5話說u
34、ni53EFuni7528 Bernstein uni591Auni9805式來逼uni8FD1而uni9019uni6B63是 Bernstein uni7684想uni6CD5。我們uni770Buni7279殊uni7684uni51FD數f(x) = ex, uni5247Bn(x;ex)=nsummationdisplayk=0parenleftBiggnkparenrightBiggeknxk(1x)nk=nsummationdisplayk=0parenleftBiggnkparenrightBigg(e1nx)k(1x)nk=bracketleftBige1nx+ 1xbrac
35、ketrightBign=bracketleftBig1 + (e1n 1)xbracketrightBignbracketleftbigg1+xnbracketrightbiggn我們有uni8DB3夠uni7406uni7531uni76F8信limnBn(x;ex) = limn(1 + xn)n = ex2.2 uni5B9Auni7406之uni8B49明我們先uni76F4接uni76F8uni6E1Buni770Buni770B:|f(x)Bn(x;f)|=vextendsinglevextendsinglevextendsinglevextendsinglevextendsin
36、glef(x)nsummationdisplayk=0parenleftBiggnkparenrightBiggf(kn)xk(1x)nkvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle=vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglensummationdisplayk=0parenleftBiggnkparenrightBiggbracketleftBiggf(x)f(kn)bracketrightBiggxk(1x)nkvex
37、tendsinglevextendsinglevextendsinglevextendsinglevextendsingle(2.7)uni5C0D (2.7) 式而言, 我們uni671F待絕uni5C0D值會趨uni8FD1於0, 乍uni770B之下似乎 |f(x) f(kn)| uni9019uni9805uni5C31夠了, 但uni9019uni53EA是假uni8C61, uni56E0uni70BAuni9084有uni53E6uni4E00個困uni96E3uni5C31是(2.7) uni7684uni53F3式是有uni9650uni9805uni548C, uni901
38、9兩個困uni96E3uni53EF分uni5225uni7531uni4E00致有uni754C (uniformly bounded) 與uni4E00致uni9023續 (uniformly continuous) 來克uni670D。 uni56E0uni70BAf C0,1, 而 0,1是uni4E00緊緻uni96C6, uni56E0uni6B64 funi5728 0,1是有uni754C且uni4E00致uni9023續, 若|f(x)| 0, () vextendsinglevextendsinglevextendsinglevextendsinglevextendsing
39、lef(x)f(kn)vextendsinglevextendsinglevextendsinglevextendsinglevextendsingle M2(), uni5247uni7531(2.11)-(2.12) uni53EF得|f(x)Bn(x;f)| 0, 0, 0 0, (0,1)limnintegraltext1 (1x2)ndxintegraltext10 (1x2)ndx=0,limnintegraltext0 (1x2)ndxintegraltext10 (1x2)ndx=1uni8B49明: 我們作個簡uni55AEuni7684估計integraldisplay 1
40、(1x2)ndxintegraldisplay 10(1x)ndx= 1n+ 1,但 limn(n+ 1)(12)n = 0 故得uni8B49。3.2 uni8936積convolution之uni5F15uni9032uni5982uni679C作個平移考慮uni51FD數 n(x ), 0,1, uni5247uni7531 n 之uni7279uni8CEAuni53EFuni77E5uni6B64時uni53EA是uni5C07山uni5CF0uni7531x = 0 uni7684位置平移至x = 。所以乘積f(x)n(x) uni5DEE不uni591Auni5C31是f() 乘
41、上山uni5CF0uni7684uni9AD8度再uni5229uni7528(3.2) 平uni5747uni4E00下我們uni53EF以uni731Cuni6E2C新uni7684uni51FD數Pn() integraldisplay 10f(x)n(x)dx=cnintegraldisplay 10f(x)1(x)2ndx (3.6)與 f() 是uni975E常接uni8FD1。uni9019uni5C31是 Landau uni7684想uni6CD5, 也是 Weierstrass uni7684想uni6CD5。 uni9019兒所uni5F15uni9032uni7684u
42、ni5C31是uni8936積 (convolution) uni7684uni6982念; uni5728數uni5B78分析中是uni975E常uni91CD要uni7684工具。uni5982uni679Cuni5C07uni56E0式 1 (x )2n uni7528uni4E8Cuni9805式uni5B9Auni7406展uni958B, 其uni5BE6uni5B83是 uni7684 2n uni6B21uni591Auni9805式; 所以uni53EFuni7528uni591Auni9805式 Pn() 來逼uni8FD1uni9023續uni51FD數f()。uni5B
43、9Auni7406uni8B49明: 我們所uni9700要uni7684是作估計; uni770BPn()與f()兩者uni9593uni7684uni8AA4uni5DEE;uni5229uni7528uni4E09角不等式|Pn()f()vextendsinglevextendsinglevextendsingle |integraldisplay 10f(x)n(x)dxintegraldisplay f(x)n(x)dxvextendsinglevextendsinglevextendsingle+vextendsinglevextendsinglevextendsingleinte
44、graldisplay +f(x)n(x)dxintegraldisplay +f()n(x)dxvextendsinglevextendsinglevextendsingle+vextendsinglevextendsinglevextendsinglef()integraldisplay +n(x)dxf()vextendsinglevextendsinglevextendsingle (3.7)uni6B63uni5982 Bernstein uni7684uni8B49明方uni6CD5, 我們uni9084是要uni7528uni5230uni4E00致有uni754C與uni4E0
45、0致uni9023續: 若|f(x)| 0 uni53D6|x| 0uni9019uni544A訴我們 uni51FD數uni53EF視uni70BA Heaviside uni51FD數H(x) 之微分(x) = ddxH(x), H(x) =0, x 0,uni56E0uni6B64 (3.16) uni53EFuni53E6uni5916表uni793Auni70BA Stieltjes 積分integraldisplay (x)(x)dx=integraldisplay (x)dH(x) = (0) (3.17)uni53E6uni5916uni7531 (3.16) 再uni52A0
46、上平移uni53EF得f(x) =integraldisplay f()(x)d = f (x)(3.18)所以 uni51FD數是乘uni6CD5uni55AE位元uni7D20 (uni5728uni8936積uni7684意義下)。uni53D6uni7279殊uni7684f(x) = eiwx、ext uni5247integraldisplay eiw(x)dx= eiwx,integraldisplay et(x)dx= ext令 x = 0 uni5247 (3.19) uni544A訴我們uni51FD數uni7684Fourier uni8B8Auni63DB與 Lapla
47、ce uni8B8Auni63DB等於1;L(x) = F(x) = 1uni9019再uni4E00uni6B21uni5370uni8B49 uni51FD數所扮uni6F14uni7684角色是乘uni6CD5uni55AE位元uni7D20。uni8A3B: 以uni8936積uni70BA乘uni6CD5uni7684代數結構是uni6CDBuni51FD分析uni4E00uni91CD要分枝, 稱uni70BA算uni5B50uni7406論, 其中最著uni540Duni7684有 Banach 代數 、 C 代數與 Von Neu-mann 代數。最常見uni7684Dirac
48、 uni5E8F列是常態分uni914D(nor-mal distribution):n(x) =radicalbiggnenx2,integraldisplay n(x)dx = 1uni56E0uni6B64廣義uni51FD數論也稱uni70BA荷佈uni7406論 (distribu-tion theory)。 uni7531 n(x) 之uni7279uni8CEAuni53EFuni77E5uni7576n很uni5927時幾乎uni90FDuni96C6中uni57280uni9EDE而uni5916uni90E8uni70BA0, 所以uni53EFuni5C0D任意uni9023續uni51FD數(x) uni57280uni9EDEuni9644uni8FD1uni53EF視uni70BA常數limnintegraldisplay n(x)(x)dxWeierstrass 逼uni8FD1uni5B9Auni7406 21integraldisplay n(x)(0)dx=(0)integraldisplay n(x)dx = (0)3.7 uni5F15uni7406: n(x) w(x)。uni8B49明: uni9019uni76F8uni7576等於integraldisplay (x)(x)dx limnin
