Complex functions of a real variable

### Introduction

You may be used to dealing with functions of a variable such as $y = f(x)$, where $x$ is called the independent variable and $y$ is called the dependent variable and typically, $y$ takes real values when $x$ takes real values. In this course, we will be interested in complex functions of a real variable. Often the real variable will represent time or frequency. Such a function, normally denoted as $x(t)$ or $X(\omega)$ is a function which takes a complex value for every real value of the independent variable $t$ or $\omega$. Pay attention to the notation carefully - $t$ or $\omega$ now becomes the independent variable and $x(t)$ or $X(\omega)$ now becomes the dependent variable. We can also think of the complex function as the combination of two real functions of the independent variable, one for the real part of $x(t)$ and one for the imaginary part of $x(t)$.

When dealing with real functions of a real variable, you may be used to plotting the function $x(t)$ as a function of $t$. However, when $x(t)$ is a complex function, there is a problem in plotting this function since for every value of $t$, we need to plot a complex number. In this case, we do one of two things - either we plot the real part of $x(t)$ versus $t$ and plot the imaginary part of $x(t)$ versus $t$, or we plot $|x(t)|$ versus $t$ and $\mbox{arg}(x(t))$ versus $t$. Either of these is fine, but we do need two plots to effectively understand how $x(t)$ changes with $t$.