Functions of complex variables, where do we use them and why

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matrixofdynamism

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This question is not about complex numbers, I know they are used for AC circuits. This question is to understand where and why we need to use Complex functions and Complex analysis? I have seen quite pretty graphs of such functions but I am pretty sure than these functions are used for much more than some mathematical artwork. The question is, what is that something?
This is since we can do calculus with such functions also, but where do we need/use them in Engineering or Physics anyway?
 

Fourier transforms, for one. Laplace transforms for another.
 

The best reason/example I can remember is Electro-Magnetics in AC circuits.
The Real Energy is "used" by the system (Resistors, Transistors, ect...) and
The Imaginary Energy is "Stored" in the Magnetic & Electric fields of the Inductor & Caps.

Hope this helps.
 

Complex variables are used to represent storage devices (inductors and capacitors) in frquency domain. Also this can be seen in reactive power. But in practice it only represents an amount of phase shift that occurs in a property. Eg 90 for an inductor and -90 for a capacitor. Also the complex variables are used in analysis to avoid deadlock eg the square root of a negative number... This cannot be evaluated unless we use a complex variable
 

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