Introduction: The complex variable

A complex number $\alpha$ is an expression of the form $\alpha=a+i b$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, satisfying the equation

$ i^{2}=-1. \tag{1.1} $

We speak of $a$ as the real part of $\alpha$ and of $b$ as the imaginary part of $\alpha$, and write

$ \alpha=a+i b: \quad a=\Re(\alpha), \quad b=\Im(\alpha). \tag{1.2} $

In the same way we may consider a complex variable $z=x+i y$, where $x$ and $y$ are real variables.

Geometrically, it is convenient to represent a complex quantity $x+i y$ by the point $(x, y)$ in a rectangular coordinate system known as the complex plane (Fig. 1.1). Thus the real numbers are all located on the $x$-axis in this plane, whereas the pure imaginary numbers (with zero real parts) are located on the $y$-axis. For some purposes it is convenient to think of the complex number $z$ as a vector in the complex plane from the origin ($x=y=0$) to the point $(x, y)$. We speak of the length of this vector as the absolute value of $z$, or as the modulus of $z$, and denote this number by $|z|$,

$ |z|=|x+i y|=\sqrt{x^{2}+y^{2}}. \tag{1.3} $

The number $x-i y$ is called the conjugate of the number $z=x+i y$ and is denoted by the symbol $\bar{z}$,

$ \bar{z}=x-i y. \tag{1.4} $

We notice that $|z|=|\bar{z}|$ and, further, that

$ z \bar{z}=(x+i y)(x-i y)=x^{2}-i^{2} y^{2}=x^{2}+y^{2}=|z|^{2}=|\bar{z}|^{2}. \tag{1.5} $

That is, the product of a complex number and its conjugate is a non-negative real number equal to the square of the absolute value of the complex number.

Addition, subtraction, multiplication, and division of complex numbers are accomplished according to the rules governing real numbers, if one writes $i^{2}=-1$, in accordance with (1.1). Thus, if

$ z_{1}=x_{1}+i y_{1}, \quad z_{2}=x_{2}+i y_{2}, \tag{1.6} $

there follows

$ \left.\begin{array}{c} z_{1}+z_{2} = (x_{1}+x_2)+i(y_{1}+y_{2}) \\ z_{1}-z_{2} = (x_{1}-x_{2})+i(y_{1}-y_{2}) \end{array}\right\}, \tag{1.7a,b} $

$ z_1 z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1 x_2 - y_1 y_2) + i(x_1y_2 + x_2y_1), \tag{1.7c} $

and also, using $z\bar{z}=|z|^2$ (1.5),

$ \begin{aligned} \frac{z_{2}}{z_{1}} &= \frac{x_{2}+i y_{2}}{x_{1}+i y_{1}}\\ &= \frac{(x_{2}+i y_{2})(x_{1}-i y_{1})}{x_{1}^{2}+y_{1}^{2}}\\ &= \bigg(\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{1}^{2}+y_{1}^{2}}\bigg)+i\bigg(\frac{x_{1} y_{2}-x_{2} y_{1}}{x_{1}^{2}+y_{1}^{2}}\bigg). \end{aligned} \tag{1.7d} $

We notice that two complex numbers are equal if and only if their real and imaginary parts are respectively equal; that is,

$ x_{1}+i y_{1}=x_{2}+i y_{2} \quad \text{ implies } \quad x_{1}=x_{2}, \quad y_{1}=y_{2}. \tag{1.8} $

In particular, a complex number is zero if and only if its real and imaginary parts are both zero.

If we introduce polar coordinates $(r, \theta)$, such that

$ x=r \cos \theta, \quad y=r \sin \theta \quad (r \geq 0), $

the complex number $z$ can be written in the polar form

$ z=x+i y=r(\cos \theta+i \sin \theta), \tag{1.9} $

where $r$ is the modulus of $z$. It should be noticed that for a given complex number the angle $\theta$ can be taken in infinitely many ways. With the convention that the modulus $r$ be non-negative, the various possible values of $\theta$ differ by integral multiples of $2\pi$. Any one of these angles is known as the argument or amplitude of $z$.

If we notice that addition or subtraction of complex numbers follows the parallelogram law of vector combination, the truth of the useful inequalities

$ |z_{1}|-|z_{2}| \leq |z_{1}+z_{2}| \leq |z_{1}|+|z_{2}| \tag{1.10} $

follows directly from elementary geometrical considerations (Fig. 1.2).