Motivation of Fourier Coefficents What motivates the definition of the definition of the Fourier coefficients? I understand that one would try to find Fourier coefficients, but not why they are defined the way they are. 
 A: The original motivation came from solving the problem of a vibrating string stretched between fixed endpoints, which arose in the mid to late 1700s. In the process of studying this equation, Mathematicians discovered standing wave solutions vibrating at frequencies that were integer multiples of a base frequency. (This is origin of harmonic analysis.) In order to solve a general problem with a more general initial displacement of the string using sums of such standing wave solutions, it became necessary to determine the coefficients for those solutions in a general sum
Clairaut and Euler discovered a way to isolate the coefficients through the amazing discovery of the orthogonality conditions for trigonometric functions. Most Mathematicians at the time thought that such a general expansion of initial displacement of the string into trigonometric functions would not be possible. So, there was a consensus that this represented a restriction on the problems that could solved, and not a way to find general solution of the vibrating string problem.
Fourier became convinced that a general "mechanical" function representing the initial position of the string would have to be expandable in such a way. Because of the work of Fourier in this direction, the coefficients were named after Fourier, even though Fourier did not discover them. Fourier extensively studied trigonometric expansions.
