Complex Analysis - Lecture Notes for Complex Analysis Frank...

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Lecture Notes for Complex Analysis Frank Neubrander Fall 2003 Analysis does not owe its really signifcant successes oF the last century to any mysterious use oF - 1, but to the quite natural circumstance that one has infnitely more Freedom oF mathematical movement iF he lets quantities vary in a plane instead oF only on a line. Leopold Kronecker Recommended Readings: 1. Walter Rudin, Real and Complex Analysis (paperback), McGraw-Hill Publishing Co., 1987 2. John B. Conway, Functions of One Complex Variable , Springer Verlag, 1986 3. Jerold E. Marsden, Michael J. Hoffman, Basic Complex Analysis , ±reeman, 1987 4. Reinhold Remmert, Theory of Complex Functions , Springer Verlag, 1991 5. E.C. Titchmarsh, The Theory of Functions , OxFord University Press, 1975 6. Joseph Bak, Donald J. Newman, Complex Analysis , Second Edition, Springer-Verlag New York, 1996 1
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2 Tentative Table of Contents CHAPTER 1: THE BASICS 1.1 The Field of Complex Numbers 1.2 Analytic Functions 1.3 The Complex Exponential 1.4 The Cauchy-Riemann Theorem 1.5 Contour Integrals CHAPTER 2: THE WORKS 2.1 Antiderivatives 2.2 Cauchy’s Theorem 2.3 Cauchy’s Integral Formula 2.4 Cauchy’s Theorem for Chains 2.5 Principles of Linear Analysis 2.6 Cauchy’s Theorem for Vector-Valued Analytic Functions 2.7 Power Series 2.8 Resolvents and the Dunford Functional Calculus 2.9 The Maximum Principle 2.10 Laurent’s Series and Isolated Singularities 2.11 Residue Calculus CHAPTER 3: THE BENEFITS 3.1 Norm-Continuous Semigroups 3.2 Laplace Transforms 3.3 Strongly Continuous Semigroups 3.4 Tauberian Theorems 3.5 The Prime Number Theorem 3.6 Asymptotic Analysis and Formal Power Series 3.7 Asymptotic Laplace Transforms 3.8 Convolution, Operational Calculus and Generalized Functions 3.9 - 3.18 Selected Topics
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Chapter 1 The Basics 1.1 The Field of Complex Numbers The two dimensional R -vector space R 2 of ordered pairs z =( x,y ) of real numbers with multiplication ( x 1 ,y 1 )( x 2 2 ):=( x 1 x 2 - y 1 y 2 ,x 1 y 2 + x 2 y 1 ) is a commutative Feld denoted by C . We identify a real number x with the complex number ( x, 0). Via this identiFcation C becomes a Feld extension of R with the unit element 1 := (1 , 0) C . We further deFne i := (0 , 1) C . Evidently, we have that i 2 - 1 , 0) = - 1. The number i is often called the imaginary unit of C although nowadays it is hard to see anything imaginary in the plane point (0 , 1). 1 Every z C admits a unique representation z )= x (1 , 0) + (0 , 1) y = x + iy, where x is called the real part of z , x = Re ( z ), and y the imaginary part of z , y = Im ( z ). The number z = x - iy is the conjugate of z and | z | := ± x 2 + y 2 = z z is called the absolute value or norm. The multiplicative inverse of 0 ± = z C is given by z - 1 = 1 z = z | z | 2 . 1 The situation was different in 1545 when Girolamo Cardano introduced complex numbers in his Ars Magna only to dismiss them immediately as subtle as they are useless . In 1702 Leibnitz described the square root of - 1as that amphibian between existence and nonexistence and in 1770 Euler was still sufficiently confused to make mistakes like - 2 - 3= 6.
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Complex Analysis - Lecture Notes for Complex Analysis Frank...

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