I have a bunch of notations I like to use for linear algebra and vector calculus which are at this point scattered across various notes and posts. I wanted to consolidate them in one place so that I can refer to them later. I am pretty sure that the ideas in here are useful and that various parts of math and physics would benefit from incorporating them. However, I’m not sure I have figured out…
Some connections between things, which I have not seen elsewhere. Maybe they mean something? 1. The Baseless Logarithm Normally one writes a logarithm with a base, \(\log_b (x)\), to mean \[y = \log_b (x) \Lra b^y = x\] And then you can change the base of the logarithm with \[\log_b (x) = \frac{\log_a (x)}{\log_a(b)}\] Which follows from rearranging \(\log_a (x) = \log_a (b^{\log_b x}) = \log_b…
I have spent a lot of time trying to think intuitively about calculus, Taylor series, divergent series, and things like that. Here are a couple things I realized at some point which I would like to have written down. (This is sort of a sequel to a much more elementary post some years ago . I know a lot more now, and have also apparently gotten a lot more verbose.) Maybe they are well-known to some…
Still more investigation into why \((-\frac{1}{2})! = \sqrt{\pi}\). Now we are circling in on the real question, which is: what could it possibly mean to take a permutation of a fractional number of elements? At this point I am not recounting regular mathematics at all but instead just meandering around my own thoughts on the subject. Read on if that’s interesting to you, but be warned, this is…
Another installment in my investigations into the confusing value \(\Gamma(\frac{1}{2}) = (-\frac{1}{2})! = \sqrt{\pi}\). Investigations on n-Spheres Factorials as Multiplicative Integrals More on √π Locating the Lemniscate This time, we survey a bunch of other places that \(\Gamma(\frac{1}{2})\) shows up, in order to fill out our board of clues. There’s not a well-defined question here, really;…