Let's say one has an iron disk (or something resembling a disk). And let's say we keep the boundary of the disk a fixed temperature. So one part of the disk will be 300K and another part 350K for example. We wish to find which temperature the interior of the disk has.
When we put the temperature on the boundary of the disk, then of course, the temperature on the interior will fluctuate a bit. But eventually, the temperature will converge to a temperature distribution which will not (or hardly) fluctuate. This temperature is called the steady-state temperature. We wish to find this steady-state temperature.
The clue for doing this, is by noticing that the steady-state temperature will be an harmonic function. That is, a function [itex]\varphi:D\rightarrow \mathbb{R}[/itex] such that
[tex]\frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2}=0[/tex]
So we wish to extend the boundary of the disk to a harmonic function on the interior of the disk.
But notice that analytic functions give rise to harmonic functions! Indeed, the real part and the imaginary part of analytic functions are harmonic by the Cauchy-Riemann equations. So the question of the existence of a harmonic function can now be reduced to the existence of an analytic function. Complex variables can be used to answer that very question.
This book
https://www.amazon.com/dp/0486613887/?tag=pfamazon01-20 treats complex variables from that point-of-view.