Academic Forum Cauchy Confers with Weierstrass. Lloyd Edgar S. Moyo, Ph.D. Associate Professor of Mathematics

Transcription

1 Academic Forum - Cauchy Cofers wih Weiersrass Lloyd Edgar S Moyo PhD Associae Professor of Mahemaics Absrac We poi ou wo limiaios of usig he Cauchy Residue Theorem o evaluae a defiie iegral of a real raioal fucio of cos ad si We also exhibi a defiie iegral wih he limiaios Fially we overcome he limiaios by usig he Weiersrass subsiuio Iroducio Le R be a raioal fucio of wo real variables Cosider a real iegral By usig he subsiuio R (cos si ) d i e where i ad is a real umber i ca be show ha R (cos si ) d i R( )d () For deails see for example Pala [] p 9 or Ahlfors [] p or Crowder ad McCusey [] p 8 By he Cauchy Residue Theorem we have i where i R( )d ( i)( i) Re s( f ( ); ) Re s( f ( ); ) i f () ( ) R i are he poles of f i he ope ui dis D { C: <} ad Res (f(); ) deoes he residue of f a he pole Hece () becomes Limiaios of Formula () R (cos si ) d Re s( f ( ); ) () I his secio we poi ou wo cases for which he formula () does o apply ad we sugges a differe mehod of aac

2 Academic Forum - Case # : A leas oe pole of f lies o he ui circle { C: } Case # : The fucio R is o a eve fucio of ad oe wishes o evaluae We give a example of such a fucio f Le (cossi)d where <α< The so ha cos R (cos si ) si cos ( ) R i i i( ) ( i) i i( ) f () ( ) R i (( i) i) The poles of f are foud by solvig (( i) i) for We ge or ( i ) i Usig he Quadraic Formula he soluios o ( i) i are i ad i Hece he poles of f are i ad i Sice - i i we coclude ha he poles i ad i ui circle { C: } Sice he raioal fucio R is o eve we see ha he iegral / of f lie o he d () is a example of a real iegral which saisfies Cases ad ad so we cao use formula () o evaluae i Remedy: The Weiersrass Subsiuio I his secio we use he well-ow Weiersrass subsiuio o evaluae () Le The a ( ) for (-) cos ( ) () ad si ( ) ()

3 Academic Forum - Replacig A by i he followig double-agle-formula ad usig () ad () we have si(a) siacosa si si ( ) cos ( ) Similarly replacig A by i he followig double-agle-formula ad usig () ad () we have cos(a) cos²a - si²a cos cos²(/) - si²(/) Also from we have a ( ) a for (-) from which we obai Hece d d R si ) d (cos R( ) d (6) For deails o he Weiersrass subsiuio see for example Holder [] p 9 We are ow ready o employ he Weiersrass subsiuio o evaluae cos d si cos Firs we chage he limis of iegraio usig a(/) Whe he a(/) a() Whe he a(/) Now usig (6) we have si cos cos d d ( ) ( )( )( ) d

4 Academic Forum - By parial fracio decomposiio we have Therefore ( )( )( ) si cos cos d d ( )( )( ) ( )d [ l l l( ) 8 a ] l ( ) Coclusio If R is a raioal fucio of wo real variables he i is labor-savig if we are able o evaluae real iegrals of he form R (cos si ) d (7) usig formula () However as we have see here exis some subbor real iegrals of he α form R (cos si ) d for which formula () is of o use The good ews is ha he α Weiersrass subsiuio whips all raioal iegrals of he form R (cos si ) d io submissio! Biographical Sech Lloyd Moyo received his BEd (Sciece) i 99 from he Uiversiy of Malawi i souher Africa He received his MSc i Mahemaics from he Uiversiy of Sussex UK i 996 ad his PhD i Mahemaics from New Mexico Sae Uiversiy i 6 He joied Hederso Sae Uiversiy i Fall He is a member of he America Mahemaical Sociey ad he Mahemaical Associaio of America Refereces Ahlfors LV Complex Aalysis: A Iroducio o he Theory of Aalyic Fucios of Oe Complex Variable d ed McGraw-Hill New Yor 966 Crowder HK ad McCusey SW Topics i Higher Aalysis The MacMilla Compay New Yor 96 Holder LI Calculus wih Aalyic Geomery Wardsworh Publishig Compay Belmo Califoria 988

5 Academic Forum - Pala BP A Iroducio o Complex Fucio Theory Spriger-Verlag New Yor 99 Cocepios of Iocece ad Experiece i Blae s The Boo of Thel ad Wordsworh s Iimaios Ode Ashley Picoc Meor: Peggy Du Bailey PhD Chair ad Professor of Eglish Absrac Boh William Blae s The Boo of Thel ad William Wordsworh s Ode: Iimaios of Immoraliy from Recollecios of Early Childhood explore he effecs of leavig a perfec eligheed world ad eerig he flawed domai of ma while suggesig he pre-exisece of he huma soul Wordsworh s ode evisios he world as a imperfec bu worhwhile evirome while Blae s poem focuses o he dagers ad pais ha accompay i The auhors pai differig picures of his world of experiece by persoifyig aure employig vivid imagery of he soul before ad afer is desce ad by usig sigifica symbols i heir respecive epigraphs o reflec heir overall views These devices wor ogeher o provide he reader wih a choice of how o hadle his owledge of he oce divie bu ow uid uiverse Essay I is he obsiae quesioigs / of sese ad ouward higs ha cosiue he bul of huma life i boh William Blae s The Boo of Thel ad William Wordsworh s Ode: Iimaios of Immoraliy from Recollecios of Early Childhood (Wordsworh 9-) Quesioig he world i which hey live boh auhors explore he effecs of leavig a perfec eligheed world ad eerig he flawed domai of ma while suggesig he pre-exisece of he huma soul Wordsworh s ode evisios he world as a imperfec bu worhwhile evirome while Blae s poem focuses o he dagers ad pais ha accompay i The auhors pai differig picures of his world of experiece by persoifyig aure employig vivid imagery of he soul before ad afer is desce ad by usig sigifica symbols i heir respecive epigraphs o reflec heir overall views These devices wor ogeher o provide he reader wih a choice of how o hadle his owledge of he oce divie bu ow uid uiverse A Romaic idea of aure as somehig o oly alive bu also compleely seie is explored quie lierally i Blae s The Boo of Thel He uses persoificaio o form a dialogue bewee he ioce ye usaisfied Thel ad he oher ihabias of he Vales of Har Described as small ad lowly he Lilly aswers Thel s cries of dissaisfacio cocerig her lac of purpose i life (Blae 7) The Lilly was o help Thel as do all of Blae s persoified aural elemes crawl ou of her melacholia The Lilly s speech