Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex exponential represents a sinusoid then why can't we just write it as a $A\sin(\omega t+\varphi)$ rather that a weird exponential with $j$ in the power?
OK, lets take example of Fourier analysis and other fields of electronics where this is used. I am confused what complex exponentials actually mean and why they show up in mathematical equations in engineering.
Somebody may answer and say that Oh, the complex exponential can be visualized as a helix. Well, sure, but what does it mean when it is used in an equation containing physical quantities as it is used in engineering for example?
Best Answer
The word "real", in "real number", is a misnomer!! Don't take it literally. So is "imaginary". Imaginary numbers are just as real as real numbers are.
Conventionally the way people seem to be taught complex numbers is that they are told that $i^2=-1$, and then go on from there by using algebra. That's not how I first learned it. I was surprised when I found out it could be done that way. I was taught initially that multiplication by $i$ is $90^\circ$ counterclockwise rotation. Look at $\{i^n : n=\ldots,-3,-2,-1,0,1,2,3,\ldots\}$. You see circular motion. In an exponential function of $n$. That's the beginning of understanding of $x\mapsto e^{ix}$ as circular motion coming from an exponential function. That's a crucial basic idea from which Fourier analysis flows. And Fourier analysis is relied on heavily not only be electrical engineering (which you mention) but number theory, wave motion, heat flow, isoperimetric problems, probability, statistics, quantum physics, number theory, cryptography, and many things.
Circular motion is real.