advance_wk1 - COMPLEX ANALYSIS III MELISSA TACY Abstract These notes follow Complex Analysis III as taught in Semester 1 2015 at the University of

advance_wk1 - COMPLEX ANALYSIS III MELISSA TACY Abstract...

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COMPLEX ANALYSIS III MELISSA TACY Abstract. These notes follow Complex Analysis III as taught in Semester 1 2015 at the University of Adelaide. They are loosely based on the text Complex Analysis by Stein and Shakarchi. My thanks to Michael Hallam for his detailed proof-reading, however of course any remaining typos are the responsibility of the author. At the end of each lecture there is a list of the tutorial and assignment problems particularly related to that class. 1
2 MELISSA TACY Complex numbers were initially defined to take the square root of a negative number. That is mathematicians wanted to be able to solve equations like x 2 = - 1 which could not be solved within the system of real numbers. This addition to our number system allows us to solve any polynomial equation P ( z ) = 0 provided of course the polynomial P ( z ) is not a constant function. For this reason we say that the complex numbers, denoted C are is the algebraic closure of the real numbers R . We have a number of ways of writing a complex number z C , one of the most productive is to identify z with a pair of real numbers ( x, y ). By convention we say that x is the real part of z , denoted < ( z ) and y is the imaginary part of z , denoted = ( z ). We write z = x + iy This representation of z C allows us to identify the complex plane C with the familiar two dimensional real plane R 2 . The identification with R 2 also allows us another way to write z , in polar form. That is z = re . Figure 1. Identification of C with R We can go between the two formats with a little bit of simple trigonometry. x = r cos θ y = r sin θ Within this framework we can see the real number system as sitting within the complex system. For x R we write x = ( x, 0) so that the x -axis in R 2 is identified with the real line. We would, however, like to do more than algebra with the complex numbers. The aim of this class is to carry over the ideas of calculus from the real number system to the complex number system. To do that we first need a few things. We need to be able to add, divide and multiply complex numbers in a sensible format. Particularly we would like to be able to talk about the inverse of z (that is we need C to be a field under whatever multiplication we define). We need to define functions mapping from domains in C . To do this we first need to understand a little about the topology of C and also define what we consider to be valid domains and ranges for functions. Linked to the above point we need to understand limits and continuity. Finally we need to define exactly what we mean when we talk about a complex derivative (and it is this definition that really drives the field of complex analysis). The identification with R 2 gets us someway to defining addition, multiplication and division. From R 2 we already know how to add complex numbers and we also get a definition for scalar multiplication. That is if z 1 C and x R we can define x · z 1 easily enough. However we really want to be able to define z 1 · z 2 for any z 1 , z 2 C . Writing these

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