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M@Man · 2005-11-30 15:50:38

A common simplification technique used in theoretical physics is to take oscillatory solutions to differential equations and use trial solutions that are complex , i.e. $e^{i \theta}$ . Then, once we have the final solution to the equations, we just truncate the imaginary part of the solution, saying that the real part of the answer corresponds to the physical variables.

Can anyone justify this for me? We've always done it, but I've never been clear on why it's supposed to work.

I think it has something to do with the linearity of the differential equations we're trying to solve: suppose z=x+iy is a solution to some linear differential equation. Then, if z*=x-iy is also a solution, then z+z* = 2x is also a solution of the differential equation, and hence x is a solution. But not all differential equations we work with have z* as another solution to the differential equation. So how does this work?

Martin · 2005-12-03 07:26:34

M@Man wrote:
A common simplification technique used in theoretical physics is to take oscillatory solutions to differential equations and use trial solutions that are complex , i.e. $e^{i \theta}$ . Then, once we have the final solution to the equations, we just truncate the imaginary part of the solution, saying that the real part of the answer corresponds to the physical variables.

Can anyone justify this for me? We've always done it, but I've never been clear on why it's supposed to work.

It is based on the “phasor” representation, which makes creative use of Euler’s equation and the fact that the differential operator and the Re[] (“take the real part”) operator commute. If you are not familiar with phasors, you can check out any EE textbook on circuit analysis. For a brief introduction, take a look at this (<—click on this link).

M@Man · 2005-12-13 16:24:35

Thanks, Martin. That makes a lot of sense. I'll need to run it through a few times in practice to get comfortable with the idea myself, but I think I'm on board with the idea now. Now that you mention it, we used a similar technique to transform the Fourier series formula into a single integral formula (the Poisson integral formula, if I remember correctly) in my Applied Math course. Makes sense.

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#1 2005-11-30 15:50:38

Using compex solutions to physical problems

#2 2005-12-03 07:26:34

Re: Using compex solutions to physical problems

M@Man wrote:

#3 2005-12-13 16:24:35

Re: Using compex solutions to physical problems

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