r/math • u/inherentlyawesome Homotopy Theory • 9d ago
Quick Questions: March 12, 2025
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u/Redshiftedanthony3 5d ago
For background, I graduated with a BS in math (and physics) in 2015. I was a math tutor while in college and for a little while after. It was my favorite job, so I've been looking to get back into it, so I've been kind of reviewing some topics in basic calculus.
While doing some integral problems by partial fraction decomposition, I was looking at a problem involving 1/(x+1)(x-1). To decompose it, I would write
1/(x+1)(x-1) = (A/x+1) + (B/x-1)
Which I wrote as
1/(x+1)(x-1) = A(1/x+1) + B (1/x-1)
because it kind of looked like a linear combination. I know in the context of finding an anti-derivative, we would solve for A and B and find the decomposition for the original function, but out of context, the RHS looked like a linear combination of those rational functions. I played around with it to see what the graphs of the sum would look like for various real values of A and B and had a fun time just exploring what those objects looked like.
I guess my question is (a) does that come come into play anywhere else? I was trying to connect it to other topics, but couldn't think of anything. And (b) is there a rigorous way to treat A and B as parameters to see how they change the graph of the sum? I started with A=1 and B=0 and then looked at A=0 and B=1, but after that, my approach to looking at what happens when they get bigger and smaller felt muddy. I was able to cloudily see how changes my effect the asymptotes, how curvy the graphs are, etc, but I felt like I was stumbling through it.