Application of complex variables to physics?

[xdrgnh]

So I'm taking my complex variables class and learning about these cool powerful theorems like the Cauchy Goursat theorem. I know this all has huge application in physics however I just don't know what they are. Currently I'm only taking freshmen E@M so I know I won't be using it there. But next semester I'm taking analytical mechanics and I want to start using the math I know for my physics. So what are some application of complex variables to physics?

 

[Dickfore]

Solve for the current in an RLC circuit.

 

[xdrgnh]

Wouldn't that be solving a 2nd order differential with complex solutions to the characteristic equations or am I missing something. Because that's cool and all but that is more or less a DE problem.

 

[Robert1986]

Well, I don't know much Physics, so I can't really answer this. However, you can do lots of interesting (real) integrals using stuff from complex analysis.

 

[micromass]

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 $\varphi:D\rightarrow \mathbb{R}$ such that

$$\frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2}=0$$

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.

 

[Dickfore]

Wouldn't that be solving a 2nd order differential with complex solutions to the characteristic equations or am I missing something. Because that's cool and all but that is more or less a DE problem.

Yes, but there is one method based on Fourier transforms, which converts the DE to a polynomial equation essentially coinciding with the characteristic equation. This leads to the concept of impedance in circuits with alternating currents.

 

[xdrgnh]

My class doesn't cover Fourier transform sadly that is covered in the PDE class for which my complex variables class is a prerequisite for. That's awesome Micromass thank you very much. I'll study a lot more on those kinds of problems over the summer. I don't know much about thermo, wave and optics but over the summer I'm self studying so I can place out of the class and I would like to apply all of the math I know to the class itself. Btw wouldn't you need to solve a PDE for that problem.

 

[QuarkCharmer]

Wouldn't that be solving a 2nd order differential with complex solutions to the characteristic equations or am I missing something. Because that's cool and all but that is more or less a DE problem.

Physics is a big giant DE problem.