Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums

A simple theoretical derivation for obtaining the Johnson thermal noise formula using window-limited Fourier transforms is presented in detail for the first time, utilizing the well-known energy theorems. In the literature, a diverse range of alternative methods already exist, and the pedagogical value of the Fourier transform approach illustrates useful mathematical principles, taught at the undergraduate level, naturally highlighting a number of physical assumptions that are not always clearly dealt with. We also proceed to survey a number of misconceptions, problems, surprises, and conundrums concerning thermal noise.