19.1. INTRODUCTION

A complex number z is an ordered pair ( x, y) of real numbers and is written as

The real numbers x and y are called the real and imaginary parts of z. In Argand s diagram, the complex number z is represented by the point P( x, y). If ( r, ?) are the polar coordinates of P, then is called the modulus of z and is denoted by z . Also is called the argument of z and is denoted by arg. z. Every non-zero complex number z can be expressed as

If z = x + iy, then the complex number x iy is called the conjugate of the complex number z and is denoted by z.

Clearly,

19.2. FUNCTION OF A COMPLEX VARIABLE

If x and y are real variables, then z = x + iy is called a complex variable. If corresponding to each value of a complex variable z(= x + iy) in a given region R, there correspond one or more values of another complex variable w (= u + iv), then w is called a function of the complex variable z and is denoted by

For example, if

where z = x + iy and w =