Summer of Math Exposition

The interesting part of this submission is that many indefinite integrals involving pi can be calculated using complex integrals and Cauchy’s Residue Theorem (CRT). But this is completely standard.

The new part is the claim that complex integrals and CRT are necessary for understanding these integrals. I don’t get this claim. What exactly needs explaining that is not explained under the standard theory? The 1/(1+/x^6) example comes close to making the point that there is an amazing coincidence that needs explanation, and this point would be made stronger (in my opinion) if the convoluted algebra were actually shown.

My biggest complaint is that there are some mathematical terms (proof, corollary) that seem to be used in a general rather than formal sense. The elementary functions were never defined, and there is no standard definition. Therefore, the central thesis is not a well-defined mathematical claim. The thesis is never formally stated, but is claimed to be “proven.”

I see what you’re going for, but you’ve got to cool the language a bit to not come off as a crank.

This is a standout entry, its not afraid of delving into the complex, and is well researched. That being said some things could of used definitions, you mention “The Riemann sphere” with little explanation, however the definition could be looked up by the reader easily enough, so that is a trivial error.

I appreciate that it was written in LaTex, most entries use a website, but i must also state my bias here as I also wrote my entry in LaTex.

Overall a standout entry with a satisfying conclusion.

Low memorability and novelty. I think this audience can consumer academic papers, but that isn’t really the intent of SOME. We are seeking exposition, not just research or proof.

Not entirely sure what the underlying thrust of the paper is… maybe something like realizing all trigonometric functions, hyperbolic functions, and their inverses are various degree-2 polynomial variants of exponential and log. Maybe I feel the thesis of the paper seemed to imply something grander than the result that actually came out of it. Or, perhaps, I am spoiled by having known this exponential-trig connection for multiple decades of my life at this point. It is a beautiful result, no doubt, and showing that complex numbers are way more than just throwing the square root of -1 into things.

I feel like the Gaussian integral e^(-x^2) is actually a valid function to talk about in this context, since functional composition is one of the allowed operations in the more algebraic theory that stems from this (differential Galois theory, if you’re interested in looking that up). Pi emerging as part of the half-integer Gamma function is my own conception of what π really is. This, however, may be getting into some opinionated weeds, so probably should not factor too much into this review.

Finally, a somewhat off-putting aspect of this is that a lot of the language sounds like a humanities paper (not that I’m an expert at that, but it’s more, the kind of stuff they make fun of humanities papers for, fairly or unfairly). Ontological emergence etc. etc. which can be confusing.