Chapter 4
Elements of Complex Analysis
4.1 The Algebra of Complex Numbers
4.1.1 Basic Definitions
Complex numbers occur naturally in several areas of physics and engineer-
ing, for example, in the study of Fourier series and transforms and related
applications to signal processing and wave propagation. They also appear
in the mathematical models of quantum mechanics. Still, the original moti-
vation to introduce complex numbers was the study of roots of polynomials.
A polynomial p = p(x) in the variable x is an expression
anxn + an-ixn~x H aix + a 0.The numbers a,j are called the coefficients of the polynomial; if an ^ 0,
then n is called the degree of the polynomial. A polynomial equation
is p(x) = 0, and a root of p is, by definition, a solution of this equation.
So far, our underlying assumption was that the reader has some ba-
sic familiarity with the construction of the real numbers. At this point,
though, we will go back to the foundations of the theory of real numbers.
In the late 1800's, German mathematicians GEORG FERDINAND LUDWIG
PHILIPP CANTOR (1845-1918) and JULIUS WILHELM RICHARD DEDEKIND
(1831-1916) put the construction of the real number system on a precise
mathematical foundation by combining, in a rather sophisticated way, set
theory and analysis. A modern approach that we will outline next, is more
algebraic and leads to the complex numbers in a natural way. The remark
God made the integers and all the rest is the work of man,attributed to the German mathematician LEOPOLD KRONECKER (1823-
1891), suggests the set N = {1,2,3,...} of positive integers as the starting
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