Introduction to Complex Analysis - excerpts B.V. Shabat June 2, 2003. Complex analysis is a classic and central area of mathematics, which is studies and exploited in a range of important fields, from number theory to engineering. 1 Introduction: why study complex analysis? 2. COMPLEX ANALYSIS An Introduction to the Theory of Analytic Functions of One Complex Variable Third Edition Lars V. Ahlfors Professor of Mathematics, Emeritus Harvard University McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan The aim of this two hour introduction is 1. to show that part of complex analysis in several variables can be obtained from the one-dimensional theory essentially by replacing indices with multi-indices. Complex analysis is a branch of mathematics that involves functions of complex numbers. It provides an extremely powerful tool with an unex-pectedly large number of applications, including in number theory, applied mathematics, physics, hydrodynamics, thermodynamics, and electrical en-gineering. Chapter 1 The Holomorphic Functions We begin with the description of complex numbers and their basic algebraic properties. A complex number is an expression of the form x + iy Real and imaginary parts of complex number. Real axis, imaginary axis, purely imaginary numbers. # \$ % & ' * +,-In the rest of the chapter use. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " The aim of this two hour introduction is 1. to show that part of complex analysis in several variables can be obtained from the one-dimensional theory essentially by replacing indices with multi-indices. Real axis, imaginary axis, purely imaginary numbers. Introduction to Complex Analysis Math 3364 Fall 2020 Instructor:Dr Giles Auchmuty * * * ... Complex numbers and functions are used throughout science and engineering - despite the fact that they are often called imaginary numbers. The core idea of complex analysis is that all the basic functions that arise in calculus, ﬂrst derived as functions of a real variable, such as powers and fractional x +iy. It provides an extremely powerful tool with an unex-pectedly large number of applications, including in number theory, applied mathematics, physics, hydrodynamics, thermodynamics, and electrical en-gineering. An Introduction to Complex Analysis. Birkhäuser Verlag Basel • Boston • Berlin Introduction to Complex Analysis in Several Variables Volker Scheidemann 7490_scheidemann_titelei 15.8.2005 14:53 Uhr Seite 3 Complex analysis is a branch of mathematics that involves functions of complex numbers. Introduction to Complex Analysis was first published in 1985, and for this much-awaited second edition the text has been considerably expanded, while retaining the style of the original. It is not just that the polynomial z2 +1 has roots, but every polynomial has roots in C: Points on a complex plane. Mathematics Subject Classification (2010) e-ISBN 978-1-4614-0195-7 DOI 10.1007/978-1-4614-0195-7 Ravi P. Agarwal Department of Mathematics Sandra Pinelas Department of Mathematics Azores University Kanishka Perera Department of Mathematical Sciences Preliminaries to Complex Analysis 1 1 Complex numbers and the complex plane 1 1.1 Basic properties 1 1.2 Convergence 5 1.3 Sets in the complex plane 5 2 Functions on the complex plane 8 2.1 Continuous functions 8 2.2 Holomorphic functions 8 2.3 Power series 14 3 Integration along curves 18 4Exercises 24 Chapter 2. In fact, much more can now be said with the introduction of the square root of 1. Introduction xv Chapter 1. Points on a complex plane. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " # \$ % & ' * +,-In the rest of the chapter use. 1 2 The fundamental theorem of algebra 3 3 Analyticity 7 4 Power series 13 5 Contour integrals 16 6 Cauchy’s theorem 21 7 Consequences of Cauchy’s theorem 26 8 Zeros, poles, and the residue theorem 35 9 Meromorphic functions and the Riemann sphere 38 Equality of two complex numbers. This algebraic way of thinking about complex numbers has a name: a complex number written in the form x +iy where x and y are both real numbers is in rectangular form.