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?