acm95a_notes

acm95a_notes - ACM 95a/100a Notes Eric Tai January 25, 2007...

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ACM 95a/100a Notes Eric Tai January 25, 2007 Contents 1 Complex Variables 4 1 .1 P o la rF o rm . ........................................ 4 1 .2 Eu l e r sF o rmu la. ...................................... 4 2 Complex Functions 5 2 .1 C omp l exExpon en t ia l ................................... 5 2 .2 C l exMu l t ip l i ca t ion . ................................. 5 2 .3 C l exD iv i s ion. ..................................... 5 2 .4 T r ig on om e t r i cF un c t ion s.................................. 5 2 .5 In t eg e rP ow e r so fC l exNumb e r s ........................... 5 2 .6 In t e rR oo t l e r s............................ 6 2.7 Complex Logarithm . ................................... 6 2.8 Branches of z 7 3 Periodicity of Complex Functions 7 3 l t s .................................... 7 3 .2 In v e r s eT r e t r i c t s ............................. 8 4 Complex Functions as Mappings 8 4.1 The Riemann Surface of z 1 2 ................................ 8 4.2 Point at In±nity . 8 4 .3 S t e r eo g raph i cP ro j e c t ionIn t e rp r e t a t ........................ 9 5 Branch Cuts 9 5 .1 B ran chP o in t s........................................ 9 5.2 Stereographically Projected Branch Cuts . . . . . .................... 9 6 Limits of Complex Functions 10 6 t inu i t y.......................................... 1 0 6 .2 Th eC l e r a t e.................................. 1 0 6 .2 au ch y -R i em annEqu a t 1 1 6.2.2 Di²erentiation Rules . ............................... 1 2 6 .3 An a ly t i c i t y ..................................... 1 2 7 Complex Integration 13 1
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CONTENTS CONTENTS 8 Contour Integration 14 8 .1 C on t ou r s .......................................... 1 4 8 .2 C t rIn t eg ra l s...................................... 1 5 8 .3 BoundC t t l s.................................. 1 5 8 .4 Th eMLBound. ...................................... 1 6 9 Integration with Antiderivatives 17 10 Cauchy-Goursat Theorem 20 1 0 .1Ex t en s ion s.......................................... 2 0 11 Analyticity with Contour Integration 20 1 1 .1Mu l t ip ly -C onn e c t edDom a in s ............................... 2 1 1 1 .2D e f o rm a t iono fC t r 2 1 12 Cauchy’s Integral Formula 22 1 2 .1D e r iv a t e so fAn a t i cF un c t s............................. 2 3 1 2 .2G e l i z edC au ch yIn t lF o rmu la. .......................... 2 4 12.3 Morera’s Theorem . . ................................... 2 5 13 Harmonic Functions 25 1 3 .1H a i cC ju g a t e.................................... 2 6 14 Orthogonal Level Curves 27 15 Modeling Flow with Laplace’s Equation 27 16 Sequences 29 1 6 .1S e r i e s ............................................ 2 9 1 6 .2T a y lo rS e r i e s ........................................ 3 0 17 Uniqueness of Analytic Functions 31 18 Power Series 31 1 8 .1R a t ioT e s t.......................................... 3 2 1 8 .2P ow e e r i e sM an ipu la t ion. ................................ 3 5 19 Laurent Series 35 20 Zeros of Analytic Functions 36 21 Isolated Singularities of Analytic Functions 37 2 1 em o v ab l eS gu r i t y................................... 3 7 2 1 o l e s............................................. 3 8 2 1 .3E s s t ia lS r i t i e s ................................... 3 9 22 Non-isolated Singularities of Analytic Functions 40 2
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CONTENTS CONTENTS 23 Residue Calculus 40 2 3 .1C au ch y sR e s idu eTh eo r em . ............................... 4 1 2 3 .2C a l cu la t in gR e s e sa tP o l e s............................... 4 1 2 3 .3C a l t e s e tS imp l eP o l e s........................... 4 2 2 3 .4R e s eC a l t ionR ev i ew. 4 2 24 Evaluating Trigonometric Integrals 43 25 Improper Integrals over the Real Axis 43 26 Improper Integrals Involving Trigonometric Functions 45 2 6 .1Jo rd an sL emm a ...................................... 4 6 27 Improper Integrals with Singularities at Finite Locations 48 2 7 .1Ind en t edC on t ou r s ..................................... 4 9 28 Integrals with Branch Points 50 2 8 .1Ju s t if ca t ionF o rC y e s r ...................... 5 1 29 Winding Number 52 30 Meromorphic Functions 52 3 0 .1Th eA rgum r c ip l e.................................. 5 3 31 Analytic Continuation 54 3 1 .1An a ly t i cC t inu a t ionA lon gaC t r ......................... 5 5 32 Conformal Mapping 55 3 2 .1S l g............................................ 5 5 3 2 r i t i lP o t s .......................................
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This note was uploaded on 11/23/2009 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

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acm95a_notes - ACM 95a/100a Notes Eric Tai January 25, 2007...

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