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 is is because, in electrical engineering, 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 is piece wise smooth and has period 1, write (usual abuse of the equals sign) rather than writing it out in sines and cosines. of course, if 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: for when . There is nothing to it; easy integral. Of course, one has to demonstrate the validity of and show that the usual differentiation rules work ahead of time, but you need to do that only once.