More explanation needed with regard to the smooth approximation of a staircase function

Question

I have checked that the answer to this question is correct. There are some shortcuts taken by the author of the answer (@Milo Brandt), due to which I need more information to understand the logic of the various steps involved. The information required is as follows:

1. I need a reference to explain the usage of "exponential weighting" to resolve the Gibbs' phenomenon issue.
2. An explanation of how the sum of the "exponential weighting" of the Fourier series terms evaluates to a single expression involving $\gamma$, the log function and the sine and cosine functions of $2*pi*n*x$. Thanks for the help.