COMPLEX ANALYSIS
CONTENTS I DI?ERENTIAL CALCULUS IN THE COMPLEX PLANE C COMPLEX NUMBERS CONVERGENT SEQUENCES AND SERIES CONTINUITY COMPLEX DERIVATIVES THE CAUCHY–RIEMANN DI?ERENTIAL EQUATIONS II INTEGRAL CALCULUS IN THE COMPLEX PLANE C COMPLEX LINE INTEGRALS THE CAUCHY INTEGRAL THEOREM THE CAUCHY INTEGRAL FORMULAS III SEQUENCES AND SERIES OF ANALYTIC FUNCTIONS, THE RESIDUE THEOREM UNIFORM APPROXIMATION POWER SERIES MAPPING PROPERTIES FOR ANALYTIC FUNCTIONS SINGULARITIES OF ANALYTIC FUNCTIONS LAURENT DECOMPOSITION A APPENDIX TO III 4 AND III 5 THE RESIDUE THEOREM APPLICATIONS OF THE RESIDUE THEOREM IV CONSTRUCTION OF ANALYTIC FUNCTIONS THE GAMMA FUNCTION THE WEIERSTRASS PRODUCT FORMULA THE MITTAG–LEFFLER PARTIAL FRACTION DECOMPOSITION THE RIEMANN MAPPING THEOREM A APPENDIX : THE HOMOTOPICAL VERSION OF THE CAUCHY INTEGRAL THEOREM B APPENDIX : THE HOMOLOGICAL VERSION OF THE CAUCHY INTEGRAL THEOREM C APPENDIX : CHARACTERIZATIONS OF ELEMENTARY DOMAINS V ELLIPTIC FUNCTIONS THE LIOUVILLE THEOREMS A APPENDIX TO THE DE?NITION OF THE PERIODS LATTICE THE WEIERSTRASS ?-FUNCTION THE FIELD OF ELLIPTIC FUNCTIONS A APPENDIX TO SECT V 3 : THE TORUS AS AN ALGEBRAIC CURVE THE ADDITION THEOREM ELLIPTIC INTEGRALS ABEL’S THEOREM THE ELLIPTIC MODULAR GROUP THE MODULAR FUNCTION J VI ELLIPTIC MODULAR FORMS THE MODULAR GROUP AND ITS FUNDAMENTAL REGION THE K/12-FORMULA AND THE INJECTIVITY OF THE J-FUNCTION THE ALGEBRA OF MODULAR FORMS MODULAR FORMS AND THETA SERIES MODULAR FORMS FOR CONGRUENCE GROUPS A APPENDIX TO VI 5 : THE THETA GROUP A RING OF THETA FUNCTIONS VII ANALYTIC NUMBER THEORY SUMS OF FOUR AND EIGHT SQUARES DIRICHLET SERIES DIRICHLET SERIES WITH FUNCTIONAL EQUATIONS THE RIEMANN ?-FUNCTION AND PRIME NUMBERS THE ANALYTIC CONTINUATION OF THE ?-FUNCTION A TAUBERIAN THEOREM VIII SOLUTIONS TO THE EXERCISES VIII 1 SOLUTIONS TO THE EXERCISES OF CHAPTER I VIII 2 SOLUTIONS TO THE EXERCISES OF CHAPTER II VIII 3 SOLUTIONS TO THE EXERCISES OF CHAPTER III VIII 4 SOLUTIONS TO THE EXERCISES OF CHAPTER IV VIII 5 SOLUTIONS TO THE EXERCISES OF CHAPTER V VIII 6 SOLUTIONS TO THE EXERCISES OF CHAPTER VI VIII 7 SOLUTIONS TO THE EXERCISES OF CHAPTER VII REFERENCES INTRODUCTION THE COMPLEX NUMBERS HAVE THEIR HISTORICAL ORIGIN IN THE 16TH CENTURY WHEN THEY WERE CREATED DURING ATTEMPTS TO SOLVE ALGEBRAIC EQUATIONS G CARDANO (1545) HAS ALREADY INTRODUCED FORMAL EXPRESSIONS AS FOR INSTANCE 5±??15, IN ORDER TO EXPRESS SOLUTIONS OF QUADRATIC AND CUBIC EQUATIONS AROUND 1560 R BOMBELLI COMPUTED SYSTEMATICALLY USING SUCH EXPRESSIONS AND FOUND 4 AS A SOLUTION OF THE EQUATION X3 = 15X + 4 IN THE DISGUISED FORM 4 = 2 + ? ?121 +