x f(t) 1 + t 1 + t 1 + u k e uk du = f(0) k Γ 1
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- Edgar Parker
- 5 years ago
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1 ½ ÌÇÌÌÆ ËÇÄÍÌÁÇÆË Ì Ö Ø ÓÐÙØÓÒ ØÓ Ø ÔÐ ØÓÒ Ó ÔÖÓÐÑ ÔÔÖÒ Ò Ø ËÔØÑÖ ¾¼¼ Ù Ò Ø ØÓ Ø ÑÑÓÖÝ Ó ÂÑ ÌÓØØÒº ÌÇÌÌÆß¼½º ¾¼¼ ¾¼ ¾¾ ÈÖÓÔÓ Ý Ó ÑÒ ÈÓÓØ ÌÙÓÖ ÎÒÙ ÆØÓÒÐ ÓÐÐ ÙÖ Ø ÊÓÑÒº ÄØ H Ø ÓÖØÓÒØÖ Ó ØÖÒÐ ABC Ò ÐØ P Ø ÓÒ ÒØÖ ØÓÒ Ó Ø ÖÙÑÖÐ Ó ØÖÒÐ AHC ÛØ Ø ÒØÖÒÐ ØÓÖ Ó BACº Á X Ø ÖÙÑÒØÖ Ó ØÖÒÐ APB Ò Y Ø ÓÖØÓÒØÖ Ó ØÖÒÐ APC ÔÖÓÚ ØØ Ø ÐÒØ Ó XY ÕÙÐ ØÓ Ø ÖÙÑÖÙ Ó ØÖÒÐ ABCº ËÓÐÙØÓÒ Ý ÂÓÒ º ÀÙÚÖ ÖÒ ÈÖÖ º Ì ÔÓÒØ A B C H ÓÖÑ Ò ÓÖØÓÒØÖ Ø Ó. Ø ÖÙÑÖÐ Ó Ø ÓÙÖ ØÖÒÐ ÓÖÑ C Ú ÓÒÖÙÒØ Öº ÁÒ ÔÖØÙÐÖ ØÖÒ¹ O Ð ABC Ò AHC Ú ÕÙÐ ÖÙÑÖ. H ÛØ AC Ü Ó. ÝÑÑØÖݺ ÆÓÛ A P. P C Y Ð Ó ÓÖÑ Ò.. ÓÖØÓÒØÖ Ø ÛØ O A Ø ÖÙÑÖÐ Ó. B ØÖÒÐ APC Ò ACY ÚÒ AC Y Ü Ó ÝÑÑØÖÝ Ò P Ð ÓÒ Ø ÖÙÑÖÐ Ó ØÖÒÐ. AHC Ò Y Ð ÓÒ Ø ÖÙÑÖÐ Ó X ØÖÒÐ ABCº ÄØ O O Ø.. ÖÙÑÒØÖ Ó ØÖ¹ ÒÐ ABC AHCº ÄÒ ÑÒØ Y C Ò O X Ö ÔÖÐÐÐ ÓØ Ö ÔÖÔÒÙÐÖ ØÓ AP º ÓÒ Ö Ø ÔÖÔÒÙÐÖ ØÓÖ Ó Y C Ô Ò ØÖÓÙ O Ò ÑÒ ÖØ ÒÐ ÛØ O Xº Ì ÖÝ ÓÖÑÒ OO X Ñ ÖØ ÒÐ ÛØ Ø ÖÝ Ó PAC Ò OO X = A Ò ÑÐÖÐÝ OXO = Aº ÁØ ÓÐÐÓÛ ØØ Ø ÔÖÔÒÙÐÖ ØÓÖ Ó Y C Ð Ó Ø O X Ò ØÖÒÐ
2 ¾¼ OXO Ó Ð º ÌÙ O C Ò XY Ö ÝÑÑØÖ Ò Ø ÔÖÔÒÙÐÖ ØÓÖ Ó Y C Ò Ò O C = XY Û ÓÐÚ Ø ÔÖÓÐÑ Ò ØÖÒÐ ABC Ò APC Ú Ø Ñ ÖÙÑÖÙ º Ð Ó ÓÐÚ Ý ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒÝѹ Ò ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÈÌÊ º ÏÇÇ ÓÐ ÍÒÚÖ ØÝ Ä ÅÖ ÍË Ò Ø ÔÖÓÔÓ Öº ÌÇÌÌÆß¼¾º ¾¼¼ ¾¼ ¾¾ ÈÖÓÔÓ Ý ÇÚÙ ÙÖÙ ÑÔ ÌÙÖÞ ÐÙ ÊÓÑÒº ÄØ k Ò ÒØÖ Ò ÐØ f : [, ) R ÓÙÒ ÓÒ¹ ØÒÙÓÙ ÙÒØÓÒº Á x ÔÓ ØÚ ÖÐ ÒÙÑÖ Ò Ø ÚÐÙ Ó lim k x f(t) + t k dt º ËÓÐÙØÓÒ Ý Ô ÙÖØ Å ÓÙÖ ËÓÙØÖÒ ËØØ ÍÒÚÖ ØÝ ÂÓÔÐÒ ÅÇ Í˺ Ì Ù ØØÙØÓÒ u = t k ØÖÒ ÓÖÑ I(, k, x) = k x f(t) + t k dt ÒØÓ I(, k, x) = x k u f + u k du = k u f + u k χ(u) du k ÛÖ χ(u) = u ä, x k ç Ò χ(u) = ÓØÖÛ º Ò g (u) = f u + u k χ(u) º k ÓÖ Ü u Ò k Ø ÒÓÑÒØÓÖ Ò ÒÖ Ò ÙÒØÓÒ Ó Ó ØØ ÛØ f M Û Ú g (u) M + u k ØØ Ò ÒØÖÐ ÙÒØÓÒ ÓÑÒØ g (u)º Ý Ä Ù³ ÓÑÒØ ÓÒÚÖÒ ÌÓÖÑ lim I(, k, x) = = f() lim g (u) du = e uk du = f() k Γ k f() exp(u k ) du º
3 ¾½ Ð Ó ÓÐÚ Ý ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ Ò Ø ÔÖÓÔÓ Öº ÌÖ Û ÓÒ Ò¹ ÓÖÖØ ÓÐÙØÓÒ Ò ÓÒ ÒÓÑÔÐØ ÓÐÙØÓÒ ÙÑØغ ÌÇÌÌÆß¼ º ¾¼¼ ¾¼ ¾ ÈÖÓÔÓ Ý ÇÚÙ ÙÖÙ ÑÔ ÌÙÖÞ ÐÙ ÊÓÑÒº ÄØ f : [, ] R ÓÒØÒÙÓÙ ÙÒØÓÒº Ò Ø ÐÑØ lim f + + x x Ǒ dx... dx º ËÓÐÙØÓÒ Ý Ø ÔÖÓÔÓ Ö ÑÓ Ý Ø ØÓÖº ÄØ k ÚÒ ÔÓ ØÚ ÒØÖº ÓÖ ÒÝ ÔÓ ØÚ ÒØÖ ÐØ I = k x + + dx dx º x ÈÙØ C k = Ê tk e t dt < º Ï ÛÐÐ Ö Ø ÓÛ ØØ lim I = º ½µ ÅÒ Ø Ù ØØÙØÓÒ y i = x i i =... Ò I Û ÓØÒ I = k dy dy (y + + y ) k y y Ò Ð Ó Û Ú e t(y + +y ) t k dt C k = e u u k du C k y + + y y + + y = e u u k du C k (y + + y ) k ÀÒ I = (y + + y ) k º = k = k C k = C k t k C k s k e ty e t(y + +y ) dy dy dt y y y dy dt e sy/ y ds ÛÖ t = s º
4 ¾¾ ÄØ Ó ØØ ÆÓÛ ÓÖ ÐÐ f (s) = s k f (s) s k e s e sy/ y I = f (s) ds º C k dy y dy = s k e s Ò ØÙ f ÒØÖÐ ÓÖ ÐÐ º ËÓ Û Ò ÓÛ lim f (s) = ØÒ ½µ ÛÓÙÐ ÓÐÐÓÛ ÖÓÑ Ø Ä Ù ÓÑÒØ ÓÒÚÖÒ ÌÓÖÑ Ò ¾µº ÌÓ ÓÛ ØØ lim f (s) = ÔÙØ X = e sy/ y dy º ÌÒ < X < ÓÖ ÐÐ Ò Ý Ø ÅÓÒÓØÓÒ ÓÒÚÖÒ ÌÓÖÑ lim X = lim ÔÓÛÖ Ö ÜÔÒ ÓÒ ÓÖ l t e sy/ y dy = y dy = º (t ) (t )3 (t )4 l t = (t ) + + < t < 3 4 Ó ØØ l X (X ) = (X ) ¾µ 3 (X ) + 4 (X ) º ËÒ lim X = Ø ÙÑ Ò Ø ÓÐÙØ Ò Ò Ø ÓÚ ÕÙØÓÒ ÔÔÖÓ Ò Ø ÐÑØ º Ð Ó X = = Ù e sy/ y dy + e sy/ = e s y e sy/ s + e s/ s e sy/ y dy < y dy e sy/ y dy + e sy/ y e sy/ dy + dy = l y dy y e sy/ y dy º
5 ¾ Ò e sy/ dy y y dy = Ø ÓÐÐÓÛ ØØ ØÖ ÓÑ ÓÒ ØÒØ C Ù ØØ ÓÖ ÙÆÒØÐÝ ÐÖ ÌÙ lim (X ) = Ò l X C lim l X = lim (X ) Ò Ø Ò ØØ ÓØ ÐÑØ ÕÙÐ Ø Ñ ÜØÒ ÖÐ ÒÙÑÖ ÓÖ ÓØ ÐÑØ Ó ÒÓØ Ü Øº ÁÒ ÔÖØÙÐÖ lim X = lim e l X ÊÐÐÒ ØØ s ÔÓ ØÚ Û Ú Ø ØÑØ e sy/ (X ) = dy y e sy/ = s y s s e sy/ y e s dy º = lim e(x ) º µ e sy/ y e sy/ dy dy y dy = se s l Ò lim (X ) = º ÌÖÓÖ ÖÓÑ µ lim X lim f (s) = s k lim X = y = Ò Ó Û ØÐ ½µº ÁÒ ÔÖØÙÐÖ Ø ÑÔÐ ØØ ÓÖ ÒÝ ÔÓÐÝÒÓÑÐ p(t) = a + a t + + a m t m Û Ú lim p x + + x dx dx = a = p() º ÒÐÐÝ ÐØ f ÒÝ ÓÒØÒÙÓÙ ÙÒØÓÒ ÓÒ [, ] Ò ÐØ ǫ > ÖØÖÖݺ Ý Ø ÏÖ ØÖ ÔÔÖÓÜÑØÓÒ ÌÓÖÑ ØÖ ÔÓÐÝÒÓÑÐ p ǫ Ù ØØ f(x) p ǫ (x) ǫ ÓÖ ÐÐ x [, ]º ÀÒ L = lim + lim = lim (f p ǫ ) p ǫ (f p ǫ ) x + + x x + + x x + + dx dx dx dx x dx dx + p ǫ () º
6 ¾ < + + ÓÖ ÐÐ x... x Ò (, ] Û Ú x x ǫ (f p ǫ ) x + + x ǫ ÓÖ ÐÐ x... x Ò (, ]º ÁØ ÓÐÐÓÛ ØØ L p ǫ () ǫ Û ØÓØÖ ÛØ Ø Ø ØØ p ǫ () f() ǫ + ÑÔÐ ØØ L = f()º Ð Ó ÓÐÚ Ý ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ Ò ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ ÌÇÌÌÆß¼º ¾¼¼ ¾¼ ¾ ÈÖÓÔÓ Ý ÂÙÒ¹Ó Ó ÊÓÑÖÓ ÅÖÕÙÞ ÍÒÚÖ ÎÐÐÓÐ ÎÐÐÓÐ ËÔÒº ØØ ËÙÔÔÓ ØØ < a < b m = ab Ò m = a + b º Á x ÔÖÓÚ x m (x + m ) b(x + a) log b a a(x + b) ËÓÐÙØÓÒ Ý ÇÐÚÖ ÙÔÐ Ö ĐÙÐ ÆÊÏ ÖÑÒݺ ÓÖ x ÐØ f(x) = g(x) = h(x) = x m (x + m ) b(x + a) log b a a(x + b) x m (x + m ) º x m (x + m ) º ËÒ f g Ò h Ö «ÖÒØÐ ÓÖ x Ò f() = g() = h() Ø ÙÆ ØÓ ÓÛ ØØ f (x) g (x) h (x) ½µ ÓÖ ÐÐ x º ËÒ m m Û Ú m x + m (a + b)x + ab m x + m º ÀÒ Ó ØØ (x + m ) (x + a)(x + b) (x + m ) (x + m ) b a x + a x + b (x + m ) Ò ½µ ØÐ º
7 ¾ Ð Ó ÓÐÚ Ý Êà ÄÌ ËÒ ÂÓ ÍË ÇÊ ÈÇËÌÇÄÇÈÇÍÄÇË Å ÓÐÓÒ Ö ÊÇ ÊÊ ÄÒ ÍÒÚÖ ØÝ ÒÖ ÄÒÓÒ ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÈÍÄ ÊÃÆ ÍÒÚÖ ØÝ Ó ÌÜ ÒÙÖ Ì ÍË ÊÆÁËÇ ÂÎÁÊ ÊÁ ÈÁÌÆ ÁË ÐÚÖÞ ÙÖÓ ÈÖÓ ÓÖÓ ËÔÒ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÌÇÅ ÄÇÆ Ì ÍÒÚÖ ØÝ Ó ËÖÒØÓÒ ËÖÒØÓÒ È ÍË ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ Ò Ø ÔÖÓÔÓ Öº ÌÖ ÛÖ ØÛÓ ÒÓÑÔÐØ ÓÐÙØÓÒ ÙÑØغ ÌÇÌÌÆß¼º ¾¼¼ ¾¼ ¾ ÖÒº ÈÖÓÔÓ Ý ÅÐ ØÐÐ ÊÓÙÒ ÄØ I Ø ÒÒØÖ Ó ØÖÒÐ ABCº ÄØ Ø ÔÓÒØ A Ù ØØ AA = (cos A) AI Ò ÐØ ÔÓÒØ B Ò C Ò ÑÐÖÐݺ Ò Ø ÖÙ Ó Ø ÖÐ Ô Ò ØÖÓÙ A B Ò C Ò ÐÓØ Ø ÒØÖº Áº ËÓÐÙØÓÒ Ý Âº Ö Ö ÍÒÚÖ ØÝ Ó ÊÒ ÊÒ Ëú Ï ÐÐ ØØ Ø ÖÙÑÖÙ Ó A B C ÕÙÐ Ø ÒÖÙ Ó Ø ÓÖÒÐ ØÖÒÐ ÛÐ Ø ÖÙÑÒØÖ Ø ÓÖØÓÒØÖ Ó Ø ØÖÒÐ ÛÓ ÚÖØ Ö Ø ÔÓÒØ ÛÖ Ø ÒÖÐ Ó ABC ØÓÙ Ø º ÒÓØ Ý I a I b Ò I c Ø Ñ Ó Ø ÒÒØÖ I Ó ABC ÙÒÖ Ö ØÓÒ Ò Ø BC CA Ò AB Ö ÔØÚÐݺ Ù AI Ø A Û Ú IAI c = IAI b = Aº ÓÒ ÕÙÒØÐÝ ØØÒ X = I b I c AI Û ØØ I c XA = 9 ÑÔÐ ØØ cos A = AX AI c = AX AI ÛÒ X = A º ËÑÐÖÐÝ B = I c I a BI Ò C = I a I b CI º Ù A B Ò C Ö Ø ÑÔÓÒØ Ó Ø Ó I a I b I c Ø Ó A B C Ö ÔÖÐÐÐ ØÓ Ò Ð Ø ÐÒØ Ó Ø Ó I a I b I c º Ý Ø ÒØÓÒ Ó Ö ØÓÒ Ø ÑÔÓÒØ D Ó II a E Ó II b Ò F Ó II c Ö Ø Ø Ó Ø ÔÖÔÒÙÐÖ ÖÓÑ I ØÓ Ø Ó Ø ÚÒ ØÖÒÐ ABC Ò ÓØÖ ÛÓÖ Ø ÖÙÑÖÐ Ó DEF Ø ÒÖÐ Ó ABCº ÅÓÖÓÚÖ Ø ÐØØÓÒ ÛØ ÒØÖ I Ò ÖØÓ / Ø I a I b I c ØÓ DEF ÛÒ Ø Ó DEF Ö ÔÖÐÐÐ ØÓ Ò Ð Ø ÐÒØ Ó Ø Ó I a I b I c º Ï ÓÒÐÙ ØØ Ø ØÖÒÐ A B C Ò DEF Ö ÓÒÖÙÒØ Ó ØØ Ø ÖÙÑÖÙ Ó A B C ÕÙÐ Ø ÒÖÙ Ó ABC ÐѺ Ï ÒÓÛ Ù Ø ÔÖÐÐÐ ÓÖÖ ÔÓÒÒ Ó Ø ØÛÓ ØÖÒÐ ØÓ ÐÓØ Ø
8 ¾ ÖÙÑÒØÖ Ó A B C ÐÐ Ø Mº ËÒ D Ø ÑÔÓÒØ Ó II a Ò IB I a = IC I a = 9 DB = DC º ÌØ Ø ÔÖÔÒÙÐÖ ØÓÖ Ó B C Ô ØÖÓÙ D ÛÐÐ ØÖÓÙ Mº ÙØ Û Û ØØ B C EF Ó ØØ DM ÑÙ Ø ÔÖÔÒÙÐÖ Ð Ó ØÓ EF º ËÑÐÖÐÝ EM FD ÛÒ M Ø ÓÖØÓÒØÖ Ó DEF º ÁÁº ËÓÐÙØÓÒ Ý Ø ÔÖÓÔÓ Öº ÄØ X(σ) Ø ÖÐ ÛØ ÒØÖ X Ò ÖÙ σº ËÒ si A = r IA ÛÖ r Ø ÒÖÙ Ó ABC Ò IA = ( cos A) IA = (si A ) IA Û ØØ IA IA = r º ÌÙ A B Ò C Ö Ø ÒÚÖ Ó A B Ò C Ò Ø ÖÐ I( r) Ò Ø ÖÙÑÖÐ Ó A B C Ý M(ρ) Ø ÒÚÖ Ó Ø ÖÙÑÖÐ O(R) Ó ABCº ÁØ ÓÐÐÓÛ ØØ M(ρ) Ø Ñ Ó O(R) ÙÒÖ Ø ÓÑÓØØÝ ÛØ ÒØÖ I Ò ØÓÖ r ÛÖ p p Ø ÔÓÛÖ Ó I ÛØ Ö ÔØ ØÓ O(R)º º ØÐ Ò ÓÙÒ Ò ÖÖÒ ÐÒ ÛØ ÒÚÖ Ú ÓÑØÖÝ Ù ÀºËºÅº ÓÜØÖ ÁÒØÖÓÙØÓÒ ØÓ ÓÑØÖÝ ËØÓÒ º º ËÒ p = IO R = rr Ø ØÓÖ r R ÛÒ M Ò Ý r MI = IO Ò ρ = r R R R = rº ÓÑÑÒغ Í Ò Ð Ð ÜÔÖ ÓÒ ÓÖ R Ò r ÓÒ ÐÝ ÓØÒ ØÖÐÒÖ ÓÓÖÒØ Ó M ÖÐØÚ ØÓ ABC M(cos B+cos C, cos C+cos A, cos A+ cos B)º ÌÙ M ÔÓÒØ X 65 Ó ÐÖ ÃÑÖÐÒ³ ÒÝÐÓÔ Ó ÌÖÒÐ ÒØÖ º Ë Åغ ÅÞÒ 67 ÂÙÒ ½µ Ôº ½ ÓÖ Ø Û Ô ÁÒ ÔÖØÙ¹ ÐÖ M Ð Ó ÓÒ Ø ÐÒ ØÖÓÙ Ø ÆÐ Ò ÖÓÒÒ ÔÓÒØ Ó Ø ÓÖÒÐ ØÖÒк Ð Ó ÓÐÚ Ý ÇÊ ÈÇËÌÇÄÇÈÇÍÄÇË Å ÓÐÓÒ Ö ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ Ò ÈÌÊ º ÏÇÇ ÓÐ ÍÒÚÖ ØÝ Ä ÅÖ Í˺ ØÐÐ ÙÖØÖ ÓÑÑÒØ ØØ ÙÒÖ Ø ÒÚÖ ÓÒ Ó ËÓÐÙØÓÒ ÁÁ Ø ÒÚÖ Ó Ø Ó ØÖÒÐ ABC Ö Ø ÖÐ (IB C ) IC A ) Ò (IA B )º ËÒ Ø ØÒ ØÓ ÖÓÑ I r Û ØØ Ó Ø ÖÐ ÖÙ r Ò ÓÒØÒ Iº Ï ÖÓÒÞ Ø ÓÒ ÙÖØÓÒ Ó ØÖ ÓÒÖÙÒØ ÖÐ ÓÒØÒÒ ÓÑÑÓÒ ÔÓÒØ ÖÓÑ ÖÐÖ ÔÖÓÐÑ ¾ ½ ¼ ¾¼¼¼ ½ Ò ¾¼¼ ½ ½ ¾¼¼ ½½¹½¾ ÛÖ ÙÖØÖ ÖÖÒ Ö ÔÖÓÚº ÖÓÑ Ö ÙÐØ ØÐ ØÖ Û ØØ I Ø ÓÖØÓÒØÖ Ó A B C Ù ØØ ØÖÒÐ ÒØÖÒ ÛØ DEF Ý ÐØÙÖÒ ÓÙØ Ø ÓÑÑÓÒ ÑÔÓÒØ Ó A Ò D B Ò E C Ò F Ò M Ò I ÓÖÒ ØÓ ËÓÐÙØÓÒ Á ÓÚµ Û Ò ØØ ρ = r Ò M Ø ÓÖØÓÒØÖ Ó DEFº ÌÇÌÌÆß¼º ¾¼¼ ¾½ ¾ ÈÖÓÔÓ Ý ÐÐ ËÒ ÍÒÚÖ ØÝ Ó ÐÖÝ ÐÖÝ º ÂÑ Ò ØÖ Ó Ù ÔÐÝ ÖÓÙÒ Ó Óк Ù ÙÐ ÂÑ ÛÓÒ Ø Ñº ÁÒ Ø Ø ÚÖÝ ØÛÓ Ó ØÖ Ù Ò Ø ÓÐÐÓÛÒ Ò º ÄØ ØÖ Ù A B Ò C Ò ÐØ a i A³ ÓÖ ÓÒ ÓÐ i È ÓÖ ÐÐ i 8 Ò ÑÐÖÐÝ Ò b i Ò c i º ËØ S ab = 8 mi(a i, b i ) i=
9 ¾ Ò ÑÐÖÐÝ Ò S ac Ò S bc º ÌÒ Âѳ ØÓØÐ ÓÖ Û Ð ØÒ S ab È S ac Ò S bc º ÀÓÛÚÖ Âѳ ÓÖ Û ÑÓÖ ØÒ S abc = 8 mi(a i, b i, c i )º Âѳ ÓÖ Û 7º Ù i= ÏØ Û Ø ÑÒÑÙÑ ÔÓ Ð ÓÖ Ó ÒÝ Ó ËÓÐÙØÓÒ Ý ÇÐÚÖ ÙÔÐ Ö ĐÙÐ ÆÊÏ ÖÑÒݺ Ì ÓÐÐÓÛÒ ÜÑÔÐ ÓÛ ØØ Âѳ Ù Ò Ú ÓÖ 75 ¾ a a... a 8 b b... b 8 c c... c 8 ¾ = º Ï ÛÐÐ ÔÖÓÚ ØØ 75 Ø ÑÒÑÙÑ ÔÓ Ð ÓÖ Ó ÒÝ Ó Âѳ Ù º ÆÓØ ØØ S abc 7 ÛÖ Ó S ab S bc Ò S ac Ø Ð Ø 73º È ÄØ S a = 8 a i º Ï ÓÛ ØØ S ab < S a Ø ÔÖÓÓ Ò Ý ÓÒØÖØÓÒº i= ÙÑ ØØ S a = S ab º ÌÒ ÓÖ i 8 Û Ú ØØ a i = mi(a i, b i ) b i Ò 73 S ac = S abc 7 ÓÒØÖØÓÒº ÌÙ 74 S ab + S a º ÁØ ÖÑÒ ØÓ ÓÛ ØØ S a = 74 ÑÔÓ ¹ к Ì ÔÖÓÓ Ò Ý ÓÒØÖØÓÒº ÙÑ ØØ S a = 74º ÄØ Ù Ò Ø ÕÙÒØØ m i = mi(a i, b i, c i ) a i = a i m i S a = S a S abc b i = b i m i S a b = S ab S abc = c i = c i m i S a c = S ac S abc = 8 i= 8 i= mi(a i, b i ) mi(a i, c i ) º ÓÖ i 8 Ø Ð Ø ÓÒ Ó Ø ÒÙÑÖ mi(a i, b i ) ÓÖ mi(a i, c i ) ÞÖÓº ÀÒ mi(a i, b i ) + mi(a i, c i ) a i º ÌÖÓÖ S a b + S a c S a = S a S abc = S ab + S abc = S a b + Û ÑÔÐ ØØ S a c º ÇÒ Ø ÓØÖ Ò S a c = S ac S abc 73 7 = ÓÒØÖØÓÒº Ì ÓÑÔÐØ Ø ÔÖÓÓº Ð Ó ÓÐÚ Ý ÌÇÅ ÄÇÆ Ì ÍÒÚÖ ØÝ Ó ËÖÒØÓÒ ËÖÒØÓÒ È ÍË ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ Ò Ø ÔÖÓÔÓ Öº ÆÓØ ØØ S a = 74 ØÒ S a = S ab + Ò a i = mi(a i, b i ) b i Ð ÓÖ ÜØÐÝ ÓÒ ÒÜ i = j Ò b j = a j º Ì ÓÒØÖØÓÒ S abc S ac 73 = 7 ØÒ Ö Ý Ò ÖÙÑÒØ ÑÐÖ ØÓ Ø ØÖ ÔÖÖÔ Ó ÓÙÖ ØÙÖ ÓÐÙØÓÒº
10 ¾ ÌÇÌÌÆß¼º ¾¼¼ ¾½ ¾ ÈÖÓÔÓ Ý ÖÙ ÒÙÐ Ò ËØ Ö ÐÒ ÍÒÚÖ ØÝ Ó ËÖÚÓ ËÖÚÓ Ó Ò Ò ÀÖÞÓÚÒº ÄØ a b Ò c ÒÓÒÒØÚ ÖÐ ÒÙÑÖ Ù ØØ a +b +c = º ÈÖÓÚ ÓÖ ÔÖÓÚ ØØ µ a ab + b bc + c ca 3 3 µ a + ab + b + bc + c + ca º ÓÑÔÓ Ø Ó ÓÐÙØÓÒ Ý ÏÐØÖ ÂÒÓÙ ÍÖ ÙÐÒÒÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖ Ò Ø ÓÒ ÔÖÓÔÓ Öº Ï ÓÛ ØØ ÐÐ ÓÙÖ ÒÕÙÐØ Óк ÓØ ÒÕÙÐØ Ö ÝÐ ÝÑÑØÖ Ó ÛØÓÙØ ÐÓ Ó ÒÖÐØÝ Û Ò ÙÑ ØØ ØÖ a b c ÓÖ a c bº µ Ì ÐØ Ò ÒÕÙÐØÝ ÓÐÐÓÛ ÖÓÑ µº ÌÓ ÔÖÓÚ Ø ÖØ Ò ÒÕÙÐØÝ Û ÓÑÓÒÞ Ø Ò ÔÖÓÚ ÑÓÖ ÒÖÐÐÝ ØØ ÓÖ ÐÐ ÒÓÒÒØÚ ÖÐ ÒÙÑÖ a b c Û Ú ÓÖ ÝÐ Ì ÕÙÚÐÒØ ØÓ a a + b + c ab a + b + c ÝÐ 3 a + b + c a a + b + c ab º 4(a + b + c )(a + b + c ab) (a + b + c bc) (a + b + c ca) [3a a (3b + 7c) + a (73b + 78bc + 69c ) a 9 (5b 3 + 4b c + 8bc + 39c 3 ) + a 8 (48b b 3 c + 75b c + 96bc c 4 ) a 7 (94b 5 + 7b 4 c + 48b 3 c + 37b c bc c 5 ) + a 6 (658b 6 38b 5 c + 399b 4 c + 98b 3 c b c bc c 6 ) a 5 (86b 7 9b 6 c + 556b 5 c + 64b 4 c b 3 c b c 5 + 9bc c 7 ) + a 4 (473b 8 38b 7 c + 395b 6 c 63b 5 c b 4 c 4 58b 3 c b c 6 4bc c 8 ) a 3 (b + c )(39b 7 98b 6 c + 3b 5 c + 49b 4 c b 3 c b c 5 8bc 6 + 5c 7 ) + a (b + c )(69b 8 56b 7 c + 583b 6 c 58b 5 c b 4 c 4 53b 3 c b c 6 8bc c 8 ) a(b + c ) (7b 7 39b 6 c + 5b 5 c 4b 4 c 3 b 3 c b c 5 39bc 6 + 3c 7 ) + (b + c ) (3b 8 6b 7 c + 7b 6 c 78b 5 c 3 + 4b 4 c 4 7b 3 c 5 + 3b c 6 54bc 7 + 3c 8 )] º
11 ¾ Ï ÒÓØ Ý g(a, b, c) Ø ÒÙÑÖØÓÖ Ó Ø ÔÖÚÓÙ ÖØÓÒº Ï Ò ØÓ ÔÖÓÚ ØØ g(a, b, c) º ÁÒ Ø Ö Ø Û ÙÔÔÓ ØØ a b cº ÌÒ b = a + s Ò c = a + s + t ÛÖ s Ò t Ö ÒÓÒÒØÚ ÖÐ ÒÙÑÖ º ËÙ ØØÙØÒ Ø ÒØÓ Ø ÜÔÖ ÓÒ ÓÖ g ÝÐ g(a, a + s, a + s + t) = 576a (s + st + t ) + 6a 9 (4s s t + 4st + 36t 3 ) + 6a 8 (69s s 3 t + 36s t + 83st t 4 ) + 4a 7 (548s s 4 t + 573s 3 t + 866s t st t 5 ) + a 6 (9876s s 5 t + 7s 4 t + 46s 3 t s t st t 6 ) + a 5 (4998s s 6 t s 5 t + 846s 4 t s 3 t s t st t 7 ) + a 4 (79s s 7 t + 638s 6 t s 5 t s 4 t s 3 t s t st t 8 ) + a 3 (66s s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t st t 9 ) + a (4656s + 656s 9 t + 833s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t st t ) + a(576s s t + 9s 9 t s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t st + 8t ) + (4s 4 + 8s 3 t + 8s t + 4st 3 + t 4 )(36s s 7 t + 47s 6 t + 796s 5 t s 4 t 4 + 7s 3 t s t 6 + 3st 7 + 3t 8 ) º ÁÒ Ø ÓÒ Û Ú a c b Ó ØØ b = a + s + t Ò c = a + s ÛÖ s Ò t Ö ÒÓÒÒØÚ ÖÐ ÒÙÑÖ º Ò Ø Ö Ø Ù ØØÙØÒ Ø ÒØÓ Ø ÜÔÖ ÓÒ ÓÖ g Ò ÑÔÐÝÒ ÛØ Ø ÐÔ Ó ÓÑÔÙØÖ ÐÖ Ý Øѵ Ý ÔÓÐÝÒÓÑÐ Ò a s Ò t ÐÐ Ó ÛÓ ÓÆÒØ Ö ÒÓÒÒØÚ Û ÓÑÔÐØ Ø ÔÖÓÓ Ó ÔÖØ µº µ Ì ÐØ Ò ÒÕÙÐØÝ ÕÙÚÐÒØ ØÓ ( + ab)( + bc)( + ca) a( + bc)( + ca) + b( + ab)( + ca) + c( + ab)( + bc) Û ÙÔÓÒ ÜÔÒÒ ÓÑ + ab + ac + bc + abc(a + b + c) + a b c a + b + c + a c + ab + bc + abc(ab + ac + bc) + 3abc º Ì ÓÚ ÒÕÙÐØÝ ÓÐÐÓÛ Ý Ò Ø ØÖ ÒÕÙÐØ ÐÓÛ Ó Ø ÖÑÒ ØÓ ÔÖÓÚ Ó Ø abc(a + b + c) 3abc ½µ a b c abc(ab + bc + ca) ¾µ + ab + bc + ca a + b + c + ab + bc + ca º µ
12 ¼ Ì ÒÕÙÐØÝ ½µ ÓÐÐÓÛ ÖÓÑ (a + b + c) 3(a + b + c ) = 3 º Ð Ó Ò c Û Ú abc ab ab + bc + ca ÖÓÑ Û ÒÕÙÐØÝ ¾µ ÓÐÐÓÛ º Ä ØÐÝ Û Ú a( a)( c) + b( a)( b) + c( b)( c) Û ÝÐ µº ÌÓ ÔÖÓÚ Ø ÖØ Ò ÒÕÙÐØÝ Û ÓÑÓÒÞ Ò Ò ÔÖÓÚ ÑÓÖ ÒÖÐÐÝ ØØ a a + b + c + ab a + b + c ÝÐ ÓÖ a + b + c Ì ÕÙÚÐÒØ ØÓ ÝÐ a a + b + c + ab º 6(a + b + c )(a + b + c + ab) (a + b + c + bc) (a + b + c + ca) [a + a (b 5c) + a (6b + bc + 45c ) + a 9 (39b 3 4b c + 8bc 9c 3 ) + a 8 (9b 4 44b 3 c + 8b c 44bc c 4 ) + a 7 (46b 5 57b 4 c b 3 c 74b c 3 4bc 4 + 4c 5 ) + a 6 (4b 6 b 5 c + 35b 4 c 4b 3 c b c 4 bc 5 + 4c 6 ) + a 5 (4b 7 6b 6 c 68b 5 c 64b 4 c 3 48b 3 c 4 68b c 5 6bc c 7 ) + a 4 (97b 8 8b 7 c + 335b 6 c 96b 5 c b 4 c 4 38b 3 c b c 6 4bc 7 + 9c 8 ) a 3 (b + c )(9b 7 + b 6 c + 65b 5 c + 99b 4 c b 3 c b c 5 + bc 6 39c 7 ) + a (b + c )(45b 8 + 6b 7 c + 63b 6 c 36b 5 c b 4 c 4 b 3 c b c 6 48bc 7 + 6c 8 ) a(b + c ) (5b 7 b 6 c + 4b 5 c + 4b 4 c 3 + 4b 3 c 4 + 4b c 5 bc 6 c 7 ) + (b + c ) (b 8 + b 7 c + 39b 6 c + 34b 5 c 3 + 4b 4 c 4 + b 3 c 5 + 3b c 6 bc 7 + c 8 )] º Ï ÒÓØ Ý f(a, b, c) Ø ÒÙÑÖØÓÖ Ó Ø ÔÖÚÓÙ ÖØÓÒº Ï Ò ØÓ ÔÖÓÚ ØØ f(a, b, c) º ÁÒ Ø Ö Ø a b cº ÌÒ b = a + s Ò c = a + s + t ÛÖ s Ò t Ö ÒÓÒÒØÚ ÖÐ ÒÙÑÖ º ËÙ ØØÙØÒ Ø ÒØÓ f Û ÓØÒ
13 ½ f(a, a + s, a + s + t) = 96a (s + st + t ) + 8a 9 (478s s t + 669st + 4t 3 ) + 8a 8 (484s s 3 t + 93s t + 77st 3 + 4t 4 ) + 8a 7 (45s s 4 t + 366s 3 t s t st t 5 ) + a 6 (46776s 6 + 7s 5 t s 4 t s 3 t s t st t 6 ) + a 5 (378s s 6 t + 489s 5 t s 4 t s 3 t s t st t 7 ) + a 4 (78867s s 7 t s 6 t + 59s 5 t s 4 t s 3 t s t 6 + 5st t 8 ) + a 3 (6448s s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t 7 + 9st t 9 ) + a (496s + 73s 9 t s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t st t ) + a(3856s + 79s t s 9 t s 8 t s 7 t s 6 t s 5 t s 4 t s 3 t s t st + 7t ) + (4s 4 + 8s 3 t + 8s t + 4st 3 + t 4 )(7s s 7 t + 844s 6 t + 94s 5 t s 4 t s 3 t 5 + 6s t st 7 + t 8 ) º ÁÒ Ø ÓÒ Û Ú a c b Ó ØØ b = a + s + t Ò c = a + s ÛÖ s Ò t Ö ÒÓÒÒØÚ ÖÐ ÒÙÑÖ º Ò Ù ØØÙØÒ Ø ÒØÓ f Ò ÑÔÐÝÒ ÛØ Ø ÐÔ Ó ÓÑÔÙØÖµ ÝÐ ÔÓÐÝÒÓÑÐ Ò a s Ò t ÐÐ Ó ÛÓ ÓÆÒØ Ö ÒÓÒÒØÚ Û ÓÑÔÐØ Ø ÔÖÓÓ Ó ÔÖØ µº Ð Ó ÓÐÚ Ý ÈÍÄ ÊÃÆ ÍÒÚÖ ØÝ Ó ÌÜ ÒÙÖ Ì Í˺ ÖÒ³ ÓÐÙØÓÒ Ð Ó Ù ÓÑÔÙØÖ Ò ÒÚÓÐÚ ÜÔÖ ÓÒ ÚÒ ÐÓÒÖ ØÒ ØÓ Ó ÓÙÖ ØÙÖ ÓÐÙØÓÒº Ì ØÓÖ ÒÓ ÓØÖ ÓÐÙØÓÒ ØÓ Ó«Ö Ò ÛÓÙÐ ÒØÖ Ø Ò ÖÚÒ ÑÓÖ ÙÑÒ ÓÐÙØÓÒ ØÓ Ø ÔÖÓÐѺ ÌÇÌÌÆß¼º ¾¼¼ ¾½ ¾ ÈÖÓÔÓ Ý ÊÖ ÀÓ ÒÓ ÓÚÖÒ¹ ÑÒØ Ó Ò ÇØØÛ Çƺ ÁÒ ØÖÒÐ ABC ÙÔÔÓ ØØ AB < ACº ÄØ D Ò M Ø ÔÓÒØ ÓÒ BC ÓÖ Û AD Ø ÒÐ ØÓÖ Ò AM Ø ÑÒº ÄØ F ÓÒ AC Ó ØØ AD ÔÖÔÒÙÐÖ ØÓ DF º ÒÐÐÝ ÐØ E Ø ÒØÖ ØÓÒ Ó AM Ò DF º ÈÖÓÚ ØØ AB DE + AB DF = AC EF º ËÓÐÙØÓÒ Ý ÑÙÒ ËÛÝÐÒ Ê ÄØÚº Ò Ø ÔÓÒØ C F G ÓÒ AB Ò B G ÓÒ AC Ó ØØ BB CC FF Ò GG Ö ÐÐ ÔÖÔÒÙÐÖ ØÓ AD ÛØ M GG º Ù AB DE + AB DF = AB(DE + DF) = AB(F D + DE) = AB F E
14 ¾ Ø ÔÖÓÐÑ ÖÙ ØÓ ÔÖÓÚÒ ØØ F E EF = AC AB º Ù Ø ÐØØÓÒ ÛØ ÒØÖ A ØØ Ø E ØÓ M Ø F E ØÓ G M Ò EF ØÓ MG Ø ÐØ¹Ò Ó ½µ Ø F E EF = G M MG º Ù ACC Ò ABB Ö ÑÐÖ Ó Ð ØÖÒÐ Ø ÖØ¹Ò Ó ½µ Ø AC AB = CC BB º Ì ÔÖÓÓ ÓÒÐÙ Ý ÒÓØÒ ØØ Ò ØÖÒÐ BC C Ò CB B Ø ÑÐÒ G M Ð Ø CC ÛÐ Ø ÑÐÒ MG Ð BB Ó ØØ CC BB = G M MG º Ð Ó ÓÐÚ Ý ÇÊ ÈÇËÌÇÄÇÈÇÍÄÇË Å ÓÐÓÒ Ö ËÃÌ ÊËÄÆÁ ÍÒÚÖ ØÝ Ó ËÖÚÓ ËÖÚÓ Ó Ò Ò ÀÖÞÓÚÒ ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÀÁÈ ÍÊÌÁË Å ÓÙÖ ËÓÙØÖÒ ËØØ ÍÒÚÖ ØÝ ÂÓÔÐÒ ÅÇ ÍË ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÌÁÀÁ ÅÃÏ ÌØ Ù ØÝ Ç ÂÔÒ ÌÀÆÇË ÅÃÇË Ö À ËÓÓÐ Ó ÃÓÞÒ ÃÓÞÒ Ö ÌÁÌÍ ÎÇÆÊÍ ÓÑ Ò Ø ÊÓÑÒ Ò Ø ÔÖÓÔÓ Öº ÌÖ Û ÓÒ ÒÓÖÖØ ÙÑ ÓÒº ½µ ÌÇÌÌÆß¼º ¾¼¼ ¾½ ¾ ÈÖÓÔÓ Ý ÊÖ ÀÓ ÒÓ ÓÚÖÒ¹ ÑÒØ Ó Ò ÇØØÛ Çƺ ÄØ Ò k ÒØÖ ÛØ Ò k º ÓÒ Ö ÒÒÖ Ù Ø ØØÒ ÖÓÙÒ ÖÙÐÖ Øк ÄØ g (k) Ø ÒÙÑÖ Ó ÛÝ ØØ k Ó Ø Ù Ø Ò Ó Ò Ó ØØ ÒÓ ØÛÓ Ó Ò Ù Ø Ö ØØÒ ÒÜØ ØÓ ÓÒ ÒÓØÖº ÌÓ ÐÐÙ ØÖØ g 6 () = g 6 () = 6 g 6 () = 9 g 6 (3) = Ò g 6 (k) = ÓÖ ÐÐ k 4º ÓÖ ÐØ f (x) = k g (k)x k º ÓÖ ÜÑÔÐ f 6 (x) = +6x+9x +x 3 = (+x) +4x+x º ØÖÑÒ ÐÐ ÓÖ Û ( + x) ØÓÖ Ó f (x)º ËÓÐÙØÓÒ Ý Ô ÙÖØ Å ÓÙÖ ËÓÙØÖÒ ËØØ ÍÒÚÖ ØÝ ÂÓÔÐÒ ÅÇ Í˺ Ì ÒÙÑÖ Ó ØÖÒ ÓÒ ØÒ Ó k ÓÒ Ò k ÞÖÓ Ù ØØ ÒÓ ØÛÓ ÓÒ Ö ÒØ k k º ÌÖØÒ Ø k Ó Ò Ù Ø ÓÒ Ò
15 Ø ÓØÖ ÞÖÓ Ò ÖØÖÖÐÝ ÒÒ Ù Ø Ø ØÖØ Ó ØÖÒ Ó ÐÒØ Û Ú k + k g (k) = k k Ù Û ÑÙ Ø ÐÑÒØ ØÖÒ Ò Û Ø Ö Ø Ò Ð Ø ÐÑÒØ Ö ÓØ ÓÒ º ÌÙ Ó ØØ g (k) = = ( k )! k!( k + )! ( k )! k!( k)! ä ( k)( k + ) k(k ) ç ( k )! f (x) = x k k!( k)! k k = x k º k k k Ý ÕÙØÓÒ ºµ Ó ÓÒÖØ ÅØÑØ ÖÑ ÃÒÙØ Ò ÈØ Ò ¾ Ò ºµ f (x) = + + 4x + + 4x º ÆÓØÒ ØØ +x ØÓÖ Ó f (x) Ò ÓÒÐÝ f = Û ÐÙÐØ f + i i = + = e iπ 4 + e iπ 4 = cos π 4 º ÀÒ + x ØÓÖ Ó f (x) Ò ÓÒÐÝ cos π = ØØ Ò 4 ÓÒÐÝ = (l + ) ÛÖ l ÒÓÒÒØÚ ÒØÖº Ð Ó ÓÐÚ Ý ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒ¹ ÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÌÇÅ ÄÇÆ Ì ÍÒÚÖ ØÝ Ó ËÖÒØÓÒ ËÖÒØÓÒ È ÍË ÂÇÄ ËÀÄÇËÊ Ý Æ ÍË ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ Ò Ø ÔÖÓÔÓ Öº ÙÔÐ Ò Ø ÔÖÓÔÓ Ö ÓØ ÓÐÚ Ø ÔÖÓÐÑ Ý Ù Ò Ø ÖÙÖ Ú ÖÐØÓÒ f (x) = f (x) + xf (x)º
16 ÌÇÌÌÆß½¼º ¾¼¼ ¾¾ ¾ ÈÖÓÔÓ Ý ºÂº ËÑÒ ÐØÓÑÑÐ Ø ÆØÖÐÒ º ØÖÑÒ ÐÐ ØÖÒÐ ABC ÛÓ ÐÒØ Ö ÔÓ ØÚ ÒØÖ Ò Ù ØØ cos C = 4 5 º ËÓÐÙØÓÒ Ý ÇÐÚÖ ÙÔÐ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÑÓ Ý Ø ØÓÖº ÄØ ABC ØÖÒÐ ÛØ a = BC b = CA c = AB Ò cos C = 4 5 º Ý Ø ÄÛ Ó Ó Ò Û Ú c = a + b 8 5 ab º ½µ ÓÒÚÖ ÐÝ a b c Ö ÔÓ ØÚ ÒØÖ Ø ÝÒ ½µ ØÒ (a b) < c < (a + b) Ò a b c Ö Ø ÐÒØ Ó ØÖÒÐ Ò Ò Ý Ø ÄÛ Ó Ó Ò cos C = 4 º ÌÙ Û ÐÐ ÔÓ ØÚ ÒØÖ ÓÐÙØÓÒ Ó ½µº 5 ØÖ a ÓÖ b Ú Ð Ý 5º Ç ÖÚ ØØ (a, b, c) ÓÐÙØÓÒ ØÒ Ó (b, a, c) Ò Ø ÙÆ ØÓ Ò Ø ÓÐÙØÓÒ ÓÖ Û 5 Ú aº ÄØ a = 5dº ÌÒ ½µ ÓÑ ÓÖ c = 5d + b 8bd c = (b 4d) + (3d) º Ì Ø ÛÐйÒÓÛÒ ÈÝØÓÖÒ ÕÙØÓÒº Ï ØÒÙ ØÛÓ ½ b 4d = lm 3d = l(m ) c = l(m + )º ÁÒ Ø Û ÓÐÚ ÓÖ a b Ò c ØÓ ÓØÒ a = 5 3 l(m ) b = 4 3 l(m ) + lm c = l(m + ) º ÀÖ a b c Ö ÔÓ ØÚ ÒØÖ Ò ÓÒÐÝ 3 l(m )(m + ) l > Ò ØÖ m > ÓÖ m > º Ï ØÒÙ ØÖ Ù ½ 3 l Ò l = 3kº ÌÒ a = 5k(m ) b = 4k(m ) + 6km c = 3k(m + ) ÛÖ k > Ò ØÖ m > ÓÖ m > º ½ 3 (m ) ÛØ m = 3kº ÌÒ m = 3k( + 3k) Ò ØÙ a = 5lk(+3k) b = 4lk(+3k)+l(+3k) c = l( +6k+9k )
17 ÛÖ l > Ò ØÖ k > Ò ÓÖ k > º ½ 3 (m + ) ÛØ m + = 3kº ÌÒ m = 3k(3k ) Ò ØÙ a = 5lk(3k ) b = 4lk(3k )+l(3k ) c = l( 6k+9k ) ÛÖ l > Ò ØÖ 3k > ÓÖ 3k > º Ì ÓÑÔÐØ Ø Ö Ø º ¾ b 4d = l(m ) 3d = lm c = l(m + )º ÁÒ Ø Û ÓØÒ a = l(m ) lm b = 3 lm c = l(m + ) º ÆÓÛ a b c Ö ÔÓ ØÚ ÒØÖ Ò ÓÒÐÝ 3 lm l > Ò 3m > º ÁÒ Ø ØÖ Ù Ð Ó Ö ¾ 3 l ÛØ l = 3kº ÌÒ a = 3k(m ) + 8km b = km c = 3k(m + ) ÛÖ k > Ò 3m > º ¾ 3 m ÛØ m = 3kº ÌÒ a = l(9k ) + 8lk b = lk c = l(9k + ) ÛÖ l > Ò 9k > º ¾ 3 ÛØ = 3kº ÌÒ a = l(m 9k ) + 8lmk b = lmk c = l(m + 9k ) ÛÖ l > Ò m > º Ì ÓÑÔÐØ Ø ÓÒ Ò Ð Ø º Ì ÓÚ ÔÖÑØÖÞØÓÒ ÜÔÐØÐÝ ÝÐ ÐÐ ÖÕÙÖ ØÖÔÐ (a, b, c)º Ð Ó ÓÐÚ Ý Êà ÄÌ ËÒ ÂÓ ÍË ÊÇ ÊÊ ÄÒ ÍÒÚÖ ØÝ ÒÖ ÄÒÓÒ ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒÝÑÒ ¹ ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÄÊÌ ËÌÄÊ ÀÖÖÐÖ ËÛØÞÖÐÒ ÈÌÊ º ÏÇÇ ÓÐ ÍÒÚÖ ØÝ Ä ÅÖ ÍË Ò Ø ÔÖÓÔÓ Öº ÌÛÓ ÒÓÑÔÐØ ÓÐÙØÓÒ ÛÖ ÙÑØغ ÂÒÓÙ Ú Ø ÖÖÒ ÓÖ Ø ÛÐйÒÓÛÒ Ø ØØ ÐÐ ÈÝØÓÖÒ ØÖÒÐ Ú Ð Ú Ð Ý 3 Ò Ù Ø ØÓ ÜÔÖ ÓÐÙØÓÒ Ò ØÖÑ Ó ÈÝØÓÖÒ ØÖÔÐ º
18 ÌÇÌÌÆß½½º ¾¼¼ ¾¾ ¾ ÈÖÓÔÓ Ý ÏÐØÖ ÂÒÓÙ ÍÖ ÙÐÒÒ¹ ÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖº µ ÄØ x y Ò z ÔÓ ØÚ ÖÐ ÒÙÑÖ Ù ØØ x+y+z = º ÈÖÓÚ ØØ 8 3 x x y y z z º 9 µ º ÄØ Ò ÐØ x x... x ÔÓ ØÚ ÖÐ ÒÙÑÖ Ù ØØ x + x + + x = º ÈÖÓÚ ÓÖ ÔÖÓÚ ØØ k= xk x k º ËÓÐÙØÓÒ Ý ÐÖØ ËØÐÖ ÀÖÖÐÖ ËÛØÞÖÐÒº µ Ì ÔÐ Ó µ ÛÒ = 3º µ Ì ÒÕÙÐØÝ Ð ÓÖ = º º ÓØ ØÐÐ Ò ÙÔÐ ÔÖÓÚ Ø ÓÙÒØÖÜÑÔÐ x = 4 x = 3 4 º Ï ÙÑ 3 Ò ÔÔÐÝ Ø ÓÐÐÓÛÒ ÒÕÙÐØÝ ÔÖÓÚ Ò Ø ÊعÄØ ÓÒÚÜ ÙÒØÓÒ ÌÓÖÑ ÊĹÌÓÖѵ º Ë Îº ÖØÓ ÐÖ ÁÒÕÙÐØ ¹ ÇÐ Ò ÆÛ ÅØÓ ÁÄ ÈÙÐ Ò ÀÓÙ ÊÓÑÒ ¾¼¼º ÊÄ ÌÓÖÑ ÄØ f(u) ÙÒØÓÒ Ò ÓÒ Ò ÒØÖÚÐ Iº ËÙÔÔÓ f ÓÒÚÜ ÓÖ ØÖ u s ÓÖ u s ÓÖ ÓÑ s I Ò ÓÖ ÓÑ Ü ÔÓ ØÚ ÒØÖ f(x) + ( ) f(y) f(s) ÓÖ ÐÐ x y I Ù ØØ x + ( )y = sº ÌÒ È È k= f(x k ) f(s) ÓÖ ÐÐ x x... x I Ø ÝÒ x k = sº k= Á Û ÓÒ Ö Ø ÙÒØÓÒ f(x) = l x x ÐØ s = ØÒ Ø ÚÒ ÒÕÙÐØÝ Ø Ñ È ÓÒ I = (, ) Ò f(x k ) f(s)º k= Ï Ú ØØ f (x) = x + x(x ) Ò x f x (x) = º ÁØ ÐÝ x ( x) ÚÖ ØØ ÓÖ x (, ) x x Ò ÓÒÐÝ x Ó f (x) ÓÖ x º ÀÒ ØÓ ÔÔÐÝ Ø ÊÄ ÌÓÖÑ Ø ÙÆ ØÓ ÔÖÓÚ ØØ f(x) + ( )f(y) f ÓÖ ÐÐ x y (, ) Ù ØØ x + ( )y = º ½µ
19 ËÒ y = x Ø ÒÕÙÐØÝ ½µ Ù ÚÐÝ ÕÙÚÐÒØ ØÓ l x x + ( ) l x x x x ( ) x x x x x l x ( x) ( ) ( + x) ( ) ( ) x ( x) (3 ) ( + x) (3 ) ( ) x ( + x) ( ) 3 ( x) 3 x º ÄØ f (x) = ( + x) ( ) 3 ( x) 3 xº Ï ÛÐÐ ÓÛ ØØ f (x) ÓÖ ÐÐ x (, )º Ï Ú f (x) = ( )( + x) 3 ( ) 3 ( x) 3 + ( 3)( ) 3 ( x) 4 x f (x) = ( )( 3)( + x) 4 + ( 3)( ) 3 ( x) 4 ( 3)( 4)( ) 3 ( x) 5 x = ( )( 3)( + x) 4 + ( 3)( ) 3 ( x) 5 [ ( )x] ¾µ ÆÓØ ØØ f = + ( ) 3 3 = ( ) 4 4 ( ) 4 4 = µ Ò f = ( ) + 3 ( ) ( 3)( ) 3
20 = 3 ( ) ( ) ( 3)( ) 4 5 = 3 ( ) 4 5 [ ( ) + ( 3)] = º µ Ð Ó Û ÖÓÑ ¾µ ØØ f (x) ÓÖ < x º ÓÑÒÒ Ø ÛØ µ Ò µ Û ÓÒÐÙ ØØ f (x) ÓÖ < x º µ ÀÒ ½µ ØÖÙ ÓÖ = 3 Ò = 4º Ï ÒÓÛ ÙÑ ØØ 5º ÄØ g(x) = x( x) 3 º Ï Ò ØØ g Ö ÓÒ ä, Ò f ÒÖ ÓÒ ä, Ò ( + x) ÒÖ ÓÒ (, )º ÆÓÛ f Ý µ Ó Û ÓÒÐÙ ØØ f (x) ÓÖ ÐÐ x (, )º Ì ÓÑÔÐØ Ø ÔÖÓÓº Ð Ó ÓÐÚ Ý ÇÊ ÈÇËÌÇÄÇÈÇÍÄÇË Å ÓÐÓÒ Ö Ò ÊÇ ÊÊ ÄÒ ÍÒÚÖ ØÝ ÒÖ ÄÒÓÒº ÈÖØ µ ÓÒÐÝ Û ÓÐÚ Ý Êà ÄÌ ËÒ ÂÓ ÍË ¾ ÓÐÙØÓÒ µ ËÃÌ ÊËÄÆÁ ÍÒÚÖ ØÝ Ó ËÖÚÓ ËÖÚÓ Ó Ò Ò ÀÖÞÓÚÒ ÅÁÀÄ ÌÁÄÄ ÊÓÙÒ ÖÒ ÀÁÈ ÍÊÌÁË Å ÓÙÖ ËÓÙØÖÒ ËØØ ÍÒÚÖ ØÝ ÂÓÔÐÒ ÅÇ ÍË ÇÄÀ ÆËÀÌÆ ÄÔÞ ÖÑÒÝ ÇÄÁÎÊ ÍÈÄ Ö ĐÙÐ ÆÊÏ ÖÑÒÝ ÂÇ ÀÇÏÊ ÈÓÖØÐ ÆÅ ÍË ÌÀÆÇË ÅÃÇË Ö À ËÓÓÐ Ó ÃÓÞÒ ÃÓÞÒ Ö Ò Ø ÔÖÓÔÓ Öº ÌÛÓ ÒÓÖÖØ ÓÐÙØÓÒ ÛÖ ÙÑØغ ÌÇÌÌÆß½¾º ¾¼¼ ¾¾ ¾ ÊÓÑÒº ÈÖÓÔÓ Ý ÅÐÝ ÒÞ Ö ÓÚ ÄØ w x y Ò z ÔÓ ØÚ ÖÐ ÒÙÑÖ ÛØ w+x+y+z = wxyz Ò ÐØ Ö Ö f(x) = x x º 3 ÈÖÓÚ ØØ 3 wxy + 3 xyz + 3 yzw+ 3 zwx f(w)+f(x)+f(y)+f(z)º ËÓÐÙØÓÒ Ý ÇÐÚÖ ÙÔÐ Ö ĐÙÐ ÆÊÏ ÖÑÒݺ Ý Ø Å¹Å ÁÒÕÙÐØÝ Û Ú wxy = w z + x z + y z + wxy 4 4 z 3 Ò Ý ÝÑÑØÖÝ Ø ÝÐ ÚÖÒØ Ó Ø ÒÕÙÐØÝ Ð Ó Óк
21 Í Ò Ø ÓÚ ÒÕÙÐØÝ Ò Ø ÅßÅ ÁÒÕÙÐØÝ ÓÒ ÑÓÖ Û Ù ØØ 3 (wxy) wxy 4 3 ÝÐ ÝÐ z º ½µ Ì ÙÒØÓÒ y = g(x) = x 3 ÓÒÚ ÓÖ x > º ÀÒ Ý ÂÒ Ò³ ÒÕÙÐØÝ Û Ú f(w) = g g + Ö 4 w 3 = 3 Ö + g 4 w 3 = 4 3 º ËÑÐÖÐÝ f(x) 4 3 f(y) 4 3 Ò f(z) 4 3 Ò Ò f(w) + f(x) + f(y) + f(z) º ¾µ Ì Ö ÒÕÙÐØÝ ÓÐÐÓÛ ÑÑØÐÝ ÖÓÑ ½µ Ò ¾µº Ý Ø ÓÒØÓÒ ÓÖ ÕÙÐØÝ ØÓ ÓÐ Ò Ø ÅßÅ ÁÒÕÙÐØÝ Ò ÂÒ Ò³ ÒÕÙÐØÝ Û ØØ ÕÙÐØÝ ÓÐ Ò Ø ÚÒ ÒÕÙÐØÝ Ò ÓÒÐÝ w = x = y = z = 4 3 º Ð Ó ÓÐÚ Ý Êà ÄÌ ËÒ ÂÓ ÍË ÇÊ ÈÇËÌÇÄÇÈÇÍÄÇË Å ÓÐÓÒ Ö ËÃÌ ÊËÄÆÁ ÍÒÚÖ ØÝ Ó ËÖÚÓ ËÖÚÓ Ó Ò Ò ÀÖÞÓÚÒ ÏÄÌÀÊ ÂÆÇÍË ÍÖ ÙÐÒÒÝÑÒ ÙÑ ÁÒÒ ÖÙ Ù ØÖ ÈÌÊ º ÏÇÇ ÓÐ ÍÒÚÖ ØÝ Ä ÅÖ ÍË Ò Ø ÔÖÓÔÓ Öº Ì ØØÑÒØ Ó Ø ÔÖÓÐÑ ÓÙÐ Ú ÒÐÙ Ø ÜØÖ ÓÒØÓÒ ØØ x 3 4 y 3 4 z 3 4 Ò w 3 4º ÇØÖÛ Ø ÙÒØÓÒ f ÑÝ ÒÓØ ÖÐ ÚÐÙ ÓÖ Ò ØÒ x = y = z = Ò w = 4 ØÒ w + x + y + z = wxyz = 8 ÙØ f(x) Ò f(y) Ö ÒÓØ ÖÐ ÒÙÑÖ º
µ(, y) Computing the Möbius fun tion µ(x, x) = 1 The Möbius fun tion is de ned b y and X µ(x, t) = 0 x < y if x6t6y 3
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