Talk:Slowly varying function

Is sine or cosine slowly varying?

--130.149.114.63 (talk) 15:53, 29 October 2017 (UTC)

From the definition given, they can only be slowly varying if sin(ax)/sin(x) and cos(ax)/cos(x) converge as x tends to infinity for all a. It's fairly clear this isn't the case, just from the formulae for sin and cos of a sum. E.g

sin(ax)/sin(x) = [sin((a-1)x+x)]/sin(x)

              = [sin((a-1)x)cos(x)+cos((a-1)x)sin(x)]/sin(x)              = sin((a-1)x)sec(x)+cos((a-1)x) 

Usually this will have singularities whenever sec(x) does, preventing it from converging, so the needed limit doesn't exist.

86.30.22.26 (talk) 22:06, 9 March 2018 (UTC)

I think readers interested in slowly varying functions would benefit from an exposure to regularly varying functions, as one cannot think of one without the other, really. The seminal work of Jovan Karamata on regular variation, theory largely made popularized through the Regular Variation book by NH Bingham, CM Goldie and JL Teugels, is of importance nowadays, as a foundation to extreme value theory. If someone is interested in co-creating the page and contributing to it, please let me know on my Talk page. KolmogorovFormalism (talk) 06:46, 13 May 2024 (UTC)