COMPLEX VARIABLES WITH APPLICATIONS

By H. FIZIA

CONTENTS ALGEBRAIC AND GEOMETRIC PRELIMINARIES THE COMPLEX FIELD RECTANGULAR REPRESENTATION POLAR REPRESENTATION TOPOLOGICAL AND ANALYTIC PRELIMINARIES POINT SETS IN THE PLANE SEQUENCES COMPACTNESS STEREOGRAPHIC PROJECTION CONTINUITY BILINEAR TRANSFORMATIONS AND MAPPINGS BASIC MAPPINGS LINEAR FRACTIONAL TRANSFORMATIONS OTHER MAPPINGS ELEMENTARY FUNCTIONS THE EXPONENTIAL FUNCTION MAPPING PROPERTIES THE LOGARITHMIC FUNCTION COMPLEX EXPONENTS ANALYTIC FUNCTIONS CAUCHY–RIEMANN EQUATION ANALYTICITY HARMONIC FUNCTIONS POWER SERIES SEQUENCES REVISITED UNIFORM CONVERGENCE MACLAURIN AND TAYLOR SERIES OPERATIONS ON POWER SERIES COMPLEX INTEGRATION AND CAUCHY’S THEOREM CURVES PARAMETERIZATIONS LINE INTEGRALS CAUCHY’S THEOREM APPLICATIONS OF CAUCHY’S THEOREM CAUCHY’S INTEGRAL FORMULA CAUCHY’S INEQUALITY AND APPLICATIONS MAXIMUM MODULUS THEOREM LAURENT SERIES AND THE RESIDUE THEOREM LAURENT SERIES CLASSI?CATION OF SINGULARITIES EVALUATION OF REAL INTEGRALS ARGUMENT PRINCIPLE 10 1 COMPARISON WITH ANALYTIC FUNCTIONS 10 2 POISSON INTEGRAL FORMULA 10 3 POSITIVE HARMONIC FUNCTIONS CONFORMAL MAPPING AND THE RIEMANN MAPPING THEOREM 11 1 CONFORMAL MAPPINGS 11 2 NORMAL FAMILIES 11 3 RIEMANN MAPPING THEOREM 11 4 THE CLASS S ENTIRE AND MEROMORPHIC FUNCTIONS 12 1 IN?NITE PRODUCTS 12 2 WEIERSTRASS’ PRODUCT THEOREM 12 3 MITTAG-LE?ER THEOREM ANALYTIC CONTINUATION 13 1 BASIC CONCEPTS 13 2 SPECIAL FUNCTIONS REFERENCES AND FURTHER READING