Complex variables, and its more advanced version, complex analysis, is one of the most fascinating areas in pure and applied mathematics. It all started when mathematicians were mystified by equations that could only be solved if you could take the square roots of negative numbers. This seemed bizarre, and back then nobody could imagine that something as strange as this could have any application in the real world. Thus the term imaginary number was born and the area seemed so odd it became known as complex.
These terms have stuck around even though the theory of complex variables has found a home as a fundamental part of mathematics and has a wide range of physical applications. In mathematics, it turns out that complex variables are actually an extension of the real variables.
A student planning on becoming a professional pure or applied mathematician should definitely have a thorough grasp of complex analysis.
Perhaps the most surprising thing about complex variables is the wide range of applications it touches in physics and engineering. In many of these applications, complex variables proves to be a useful tool. For example, because of Euler’s identity, a formula we use over and over again in this book, electromagnetic fields are easier to deal with using complex variables.
Other areas where complex variables plays a role include fluid dynamics, the study of temperature, electrostatics, and in the evaluation of many real integrals of functions of a real variable.
In quantum theory, we meet the most surprising revelation about complex variables. It turns out they are not so imaginary at all. Instead, they appear to be as “real” as real numbers and even play a fundamental role in the working of physical systems at the microscopic level.