Q:

Applications of Complex Numbers

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start on my research for the course. Anything I have seen on the complex field has only been in passing from my other course like ODE, Linear Algebra, and Abstract Algebra. I was wondering if someone can lead me into the right direction about what applications of complex numbers I can look into for my research topic. Recommended books I can refer to would also help.  
Thank you for your time and thanks in advanced for your feedback. 

A:

Complex numbers are used quite extensively in electrical engineering.  Capacitors and inductors behave as resistors with a complex magnitude.  One has units like volt-amperes (reactive power), vs watts (real power).  
On the other hand, when you see in relativity, $x^2+y^2+z^2+(ict)^2$ as a distance, the $i$ does not serve the same role as the complex number, since you never find anything akin to eg $2+i$.  The relativity, and the electromagnetism it is found from, is actually based on quarterions, or ''four vectors'' rather than complex numbers.

A:

The Schrodinger Equation from quantum physics is most neatly written using complex numbers and the (complex-valued) field $\psi$
$$i \hbar \frac{\partial}{\partial t} \psi = \hat{H} \psi$$
It is possible to split this equation up into a few real-valued equations, but you lose some elegance in the process.

A:

You'd be hard pressed to find examples of real-life quantities that are better explained by Complex numbers than the Reals, but there are plenty of real-life phenomena which, although they are observed on the real number line, can be much better explained and, despite their name, simplified through the math of complex numbers. You see this a lot in Physics.

