[Physics] Can Laplace’s equation be solved using Fourier transform instead of Fourier series

electromagnetismelectrostaticsfourier transformmathematical physicssignal processing

Sorry for the long text, but I am unable to make my question more compact.

Any periodic function can be Fourier expanded. Usually, they say in mathematical physics books, if the function is not periodic we use Fourier transform which is more general than Fourier series expansion.

If Fourier transform is more general, cannot we use it to expand a periodic functions as well? Why periodic functions in textbooks are only Fourier expanded but not Fourier transformed?

More specifically, the boundary value problems that we solve in electromagnetism (like in chapter 3 of Griffiths) in which for example some potential is specified on the boundary of some region and we want to find the potential inside that region, this problem is usually solved by separation of variable then eventually applying Fourier series expansion to fit the boundary conditions. Those problems are never solved using Fourier transform, why is that? is it because that in Fourier series expansion one has control on truncating the series to whatever accuracy one wants whereas for Fourier transform one cannot do that? or is it an issue of convergence?

If both are viable there must be some criteria on using one over the other!

If one can point out a reference in which Laplace's equation is solved once with Fourier series and once with Fourier transform that will be greatly appreciated.

Best Answer

The Fourier transform of a periodic function is a delta function at every integer position with coefficient equal to the corresponding Fourier series value. You can show this by multiplying the function by a very wide Gaussian and taking the limit. The mathematical theory is made rigorous in the subject of tempered distributions.

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