FUNCTIONS OF A COMPLEX VARIABLE WITH APPLICATIONS
CONTENTS CHAPTER 1 COMPLEX NUMBERS AND THE COMPLEX PLANE COMPLEX NUMBERS AND THEIR PROPERTIES COMPLEX PLANE POLAR FORM OF COMPLEX NUMBERS POWERS AND ROOTS SETS OF POINTS IN THE COMPLEX PLANE APPLICATIONS CHAPTER 1 REVIEW QUIZ CHAPTER 2 COMPLEX FUNCTIONS AND MAPPINGS COMPLEX FUNCTIONS COMPLEX FUNCTIONS AS MAPPINGS LINEAR MAPPINGS SPECIAL POWER FUNCTIONS THE POWER FUNCTION ZN THE POWER FUNCTION Z1/N RECIPROCAL FUNCTION LIMITS AND CONTINUITY LIMITS CONTINUITY CHAPTER 2 REVIEW QUIZ CHAPTER 3 ANALYTIC FUNCTIONS DI?ERENTIABILITY AND ANALYTICITY CAUCHY-RIEMANN EQUATIONS HARMONIC FUNCTIONS CHAPTER 3 REVIEW QUIZ CHAPTER 4 ELEMENTARY FUNCTIONS EXPONENTIAL AND LOGARITHMIC FUNCTIONS COMPLEX EXPONENTIAL FUNCTION COMPLEX LOGARITHMIC FUNCTION COMPLEX POWERS TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS COMPLEX TRIGONOMETRIC FUNCTIONS COMPLEX HYPERBOLIC FUNCTIONS INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS CHAPTER 4 REVIEW QUIZ CHAPTER 5 INTEGRATION IN THE COMPLEX PLANE REAL INTEGRALS COMPLEX INTEGRALS CAUCHY-GOURSAT THEOREM INDEPENDENCE OF PATH CAUCHY’S INTEGRAL FORMULAS AND THEIR CONSEQUENCES CAUCHY’S TWO INTEGRAL FORMULAS SOME CONSEQUENCES OF THE INTEGRAL FORMULAS CHAPTER 5 REVIEW QUIZ CHAPTER 6 SERIES AND RESIDUES SEQUENCES AND SERIES TAYLOR SERIES LAURENT SERIES ZEROS AND POLES RESIDUES AND RESIDUE THEOREM SOME CONSEQUENCES OF THE RESIDUE THEOREM EVALUATION OF REAL TRIGONOMETRIC INTEGRALS EVALUATION OF REAL IMPROPER INTEGRALS INTEGRATION ALONG A BRANCH CUT THE ARGUMENT PRINCIPLE AND ROUCH´E’S THEOREM SUMMING IN?NITE SERIES CHAPTER 6 REVIEW QUIZ CHAPTER 7 CONFORMAL MAPPINGS CONFORMAL MAPPING LINEAR FRACTIONAL TRANSFORMATIONS SCHWARZ-CHRISTO?EL TRANSFORMATIONS POISSON INTEGRAL FORMULAS BOUNDARY-VALUE PROBLEMS FLUID FLOW CHAPTER 7 REVIEW QUIZ COMPLEX NUMBERS AND THE COMPLEX PLANE APPLICATIONS CHAPTER 1 REVIEW QUIZ INTRODUCTION IN ELEMENTARY COURSES YOU LEARNED ABOUT THE EXISTENCE, AND SOME OF THE PROPERTIES, OF COMPLEX NUMBERS BUT IN COURSES IN CALCULUS, IT IS MOST LIKELY THAT YOU DID NOT EVEN SEE A COM- PLEX NUMBER IN THIS TEXT WE STUDY NOTHING BUT COMPLEX NUMBERS AND THE CALCULUS OF FUNCTIONS OF A COMPLEX VARIABLE WE BEGIN WITH AN IN-DEPTH EXAMINATION OF THE ARITHMETIC AND ALGEBRA OF COMPLEX NUMBERS NO ONE PERSON “INVENTED” COMPLEX NUMBERS, BUT CONTROVERSIES SURROUNDING THE USE OF THESE NUMBERS EXISTED IN THE SIXTEENTH CENTURY IN THEIR QUEST TO SOLVE POLYNOMIAL EQUATIONS BY FORMULAS INVOLVING RADICALS, EARLY DABBLERS IN MATHEMATICS WERE FORCED TO ADMIT THAT THERE WERE OTHER KINDS OF NUMBERS BESIDES POSITIVE INTEGERS EQUATIONS SUCH AS X2 + 2X + 2 = AND X3 = 6X + 4 THAT YIELDED “SOLUTIONS” 1 + ??1 AND 2 + ??2 + COMPLEX NUMBER A COMPLEX NUMBER IS ANY NUMBER OF THE FORM Z = A + IB WHERE A AND B ARE REAL NUMBERS AND I IS THE IMAGINARY UNIT NOTE: THE IMAGINARY PART OF Z = 4 ?9I IS ?9 NOT ?9I TERMINOLOGY THE NOTATIONS A + IB AND A + BI ARE USED INTERCHANGEABLY THE REAL NUMBER A IN Z = A+ IB IS CALLED THE REAL PART OF Z; THE REAL NUMBER B IS CALLED THE IMAGINARY PART OF Z THE REAL AND IMAGINARY PARTS OF A COMPLEX NUMBER Z ARE ABBREVIATED RE(Z) AND IM(Z), RESPECTIVELY FOR EXAMPLE, IF Z = 4 ?9I, THEN RE(Z) = 4 AND IM(Z) = ?9 A REAL CONSTANT MULTIPLE OF THE IMAGINARY UNIT IS CALLED A PURE IMAGINARY NUMBER FOR EXAMPLE, Z = 6I IS A PURE IMAGINARY NUMBER TWO COMPLEX NUMBERS ARE EQUAL IF THEIR , A2 ?= 0, OR B2 ?= IN VIEW OF THESE LAWS, THERE IS NO NEED TO MEMORIZE THE DE?NITIONS OF ADDITION, SUBTRACTION, AND MULTIPLICATION