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Wiki on Functional Equations Intro

Hey guys.I recently wrote this wiki on functional equations intro. I am not entirely happy about it, and would greatly appreciate it if you could take a look, point out errors, and suggest what else should be added to the page.I am not sure whether or not to talk about things such as inverses, cyclic functions etc.. Any feedback will be appreciated.

By all means,if you can see that the note can be improved, do it.

Note by Rahul Saha
2 years, 9 months ago

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@Rahul Saha , Thanks for writing a good long entry for the Functional Equations wiki. Congrats on having such a mastery over Latex, I am certainly impressed :)

I think that it is worthwhile to add a solution to the initial list of "try to guess a solution", especially for Cauchy's functional equation \( f(x+y) = f(x) + f(y) \). This is actually a pretty surprising function to deal with, and most people would not get a solution that is not of the form \( f(x) = C x \). It would be insightful for them to understand why there could be other solutions, and how induction / rationality is related in this problem.

Calvin Lin Staff - 2 years, 9 months ago

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Thanks for the feedback.When you say"add a solution",do you mean that I should add how to solve Cauchy over the rationals,and then show that other functions can also exist over the reals if we overlook continuity?

Also,I considered doing a section called "Prerequisites" but didn't do so because I am not sure what should go in that section.Any ideas?

Rahul Saha - 2 years, 9 months ago

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Indeed. Explain why \( f(q) = q f(1) \) for rational \( q \), while we could still have \( f( \pi ) = 0 \).

What do you mean by "Prerequisites"? Is it "You should be familiar with algebraic manipulation, completing the square, etc"? If so, I think simply writing this sentence is sufficient. We can also link out to the relevant wiki pages, which they can read if they are uncertain what it means.

Calvin Lin Staff - 2 years, 9 months ago

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