Question: Fourier Transform: Why Different Scaling?

My exposure to the topic of Fourier Transforms is very cursory--maybe a few weeks work as part of an undergraduate math course (in an engineering program.) However, I am now returning to the subject for a while and, reading material, noticed that some routines scale the Fourier coefficients differently. So now I am wondering: Why do different routines scale the coefficients differently? If the purpose of the Fourier Transform is to extract the components of circular motion from data, shouldn't it all work out the same?

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