College Math Teaching

August 6, 2014

Where “j” comes from

I laughed at what was said from 30:30 to 31:05 or so:

If you are wondering why your engineering students want to use j = \sqrt{-1} is is because, in electrical engineering, i usually stands for “current”.

Though many of you know this, this lesson also gives an excellent reason to use the complex form of the Fourier series; e. g. if f is piece wise smooth and has period 1, write f(x) = \Sigma^{k = \infty}_{k=-\infty}c_k e^{i 2k\pi x} (usual abuse of the equals sign) rather than writing it out in sines and cosines. of course, \overline{c_{-k}} = c_k if f is real valued.

How is this easier? Well, when you give a demonstration as to what the coefficients have to be (assuming that the series exists to begin with, the orthogonality condition is very easy to deal with. Calculate: c_m= \int^1_0 e^{i 2k\pi t}e^{i 2m\pi x} dx for when k \ne m . There is nothing to it; easy integral. Of course, one has to demonstrate the validity of e^{ix} = cos(x) + isin(x) and show that the usual differentiation rules work ahead of time, but you need to do that only once.

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