Mathematical Methods for Physicists: A concise introduction - Site Map

6
Functions of a complex variable
The theory of functions of a complex variable is a basic part of mathematical
analysis. It provides some of the very useful mathematical tools for physicists and
engineers. In this chapter a brief introduction to complex variables is presented
which is intended to acquaint the reader with at least the rudiments of this
important subject.
Complex numbers
The number system as we know it today is a result of gradual development. The
natural numbers (positive integers 1, 2, ...) were ®rst used in counting. Negative
integers and zero (that is, 0, 1; 2; ...) then arose to permit solutions of equations
such as x ‡ 3 ˆ 2. In order to solve equations such as bx ˆ a for all integers
a and b where b 6ˆ 0, rational numbers (or fractions) were introduced. Irrational
numbers are numbers
p
which cannot be expressed as a/b, witha and b integers and
b 6ˆ 0, such as 2 ˆ 1:41423;ˆ 3:14159
Rational and irrational numbers are all real numbers. However, the real number
system is still incomplete. For example, there is no real number x which
satis®es the algebraic equation x 2 p
‡ 1 ˆ 0: x ˆ
p
1 . The problem is that we
do not know what to make of 1 because
p there is no real number whose square
is 1. Euler introduced the symbol i ˆ
1 in 1777 years later Gauss used the
notation a ‡
p
ib to denote a complex number, where a and b are real numbers.
Today, i ˆ
1 is called the unit imaginary number.
In terms of i, the answer to equation x 2 ‡ 1 ˆ 0isx ˆ i. It is postulated that i
will behave like a real number in all manipulations involving addition and multiplication.
We now introduce a general complex number, in Cartesian form
z ˆ x ‡ iy
…6:1†
233