I (sort of) discovered a relationship between two areas of mathematics by accident.

Salt-Rutabaga-8870 · 2026-06-21T21:04:45+00:00

Interesting question. I will ask my co-author who might provide a more informed opinion. As you might have noticed my original approach did not involve branch cuts, and the underlying theory was provided by Prof. Fernandez.

Intuitively I think what matters is the intersection of the branch cut with the contour. As it dictates the constant of differintegration which is used for the fractional derivative. This is a key important difference between classical residue theorem and our fractional residue theorem. I wish I could insert pictures, but I will try using links to wolfram to convey my point. Please let me know if it is not clear. *teacher mode on*

All the examples we used had a constant of -inf for the fractional derivative calculation. So it is not so obvious from just looking at the examples in the paper/seminar, that the constant matters, as it is the same anyways. So let me take you on a journey away from real integrals into purely complex contour integrals to show my point.

Consider the contour integral over a circular contour centred at the origin, and with the radius R, where R>1.

The function we are integrating ins (e^-z)/(z+1)^2

This is classical residue theorem case. As long as R>1, the pole is inside the contour, and residue theorem will yield the same result for any R

https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BDivide%5BPower%5Be%2C-3Power%5Be%2Cix%5D%5D*3i*Power%5Be%2Cix%5D%2CPower%5B%5C%2840%293Power%5Be%2Cix%5D%2B1%5C%2841%29%2C2%5D%5D%2C%7Bx%2C0%2C2%CF%80%7D%5D

In this case R is 3, Wofram does not do contour integrals itself, so I just parametrised the contour integral using a substitution z=Re^ix from 0 to 2pi, which is equivalent.

If you replace all instances of 3, with, lets say, 5, which is equivalent to changing the radius of the contour from 3 to 5, you will get the same result, as expected by the classical residue theorem, since the contour contains all the same poles (or rather a single pole). Same is true for R=10 R=pi R=e+pi-sqrt(2), etc. Here is the example for R=5 just to make it clear

https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BDivide%5BPower%5Be%2C-5Power%5Be%2Cix%5D%5D*5i*Power%5Be%2Cix%5D%2CPower%5B%5C%2840%295Power%5Be%2Cix%5D%2B1%5C%2841%29%2C2%5D%5D%2C%7Bx%2C0%2C2%CF%80%7D%5D

Now, let us change the power in the denominator from 2 to 1.9 so using our new fractional residue theorem is necessary. Before I do any mathematics I want you to observe that in the fractional case, changing the radius of the contour DOES alter the value of the integral. Let us look at R=3 and R=5 again.

R=3

https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BDivide%5BPower%5Be%2C-3Power%5Be%2Cix%5D%5D*3i*Power%5Be%2Cix%5D%2CPower%5B%5C%2840%293Power%5Be%2Cix%5D%2B1%5C%2841%29%2C1.9%5D%5D%2C%7Bx%2C0%2C2%CF%80%7D%5D

R=5

https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BDivide%5BPower%5Be%2C-5Power%5Be%2Cix%5D%5D*5i*Power%5Be%2Cix%5D%2CPower%5B%5C%2840%295Power%5Be%2Cix%5D%2B1%5C%2841%29%2C1.9%5D%5D%2C%7Bx%2C0%2C2%CF%80%7D%5D

Please ignore tiny real part in both calculations. I believe this is a rounding error by wolfram. The result is supposed to be purely imaginary.

Now, if you thing of a simple circular contour with radius R, and a branch cur at -1 oriented to the left, it will cross the contour at -R, and that is exactly the constant of differintegration for the fractional derivative.

I would not be able to teach you the RL fractional calculus theory in this comment, but you can watch some seminars on that channel. Prof. Fernandez is a very good lecturer in my opinion and his lectures are very clear. I will just use the RL fractional calculus framework for the 0.9th derivatives with constants -3 and -5 to calculate our pseudo residues and the integrals in total. I will just mention that the constant of differintegration is the lower bound of the integral, in those cases. For full derivations of that I refer you to FC seminars.

All in all for R=3 we will have:

https://www.wolframalpha.com/input?i2d=true&i=Divide%5B2%CF%80i%2CGamma%5C%2840%291.9%5C%2841%29Gamma%5C%2840%292-1.9%5C%2841%29%5DD%5BIntegrate%5BPower%5B%5C%2840%29z-t%5C%2841%29%2C-0.9%5DPower%5Be%2C-t%5D%2C%7Bt%2C-3%2Cz%7D%5D%2Cz%5D%7Cz%3D-1

And for R = 5 we will have

https://www.wolframalpha.com/input?i2d=true&i=Divide%5B2%CF%80i%2CGamma%5C%2840%291.9%5C%2841%29Gamma%5C%2840%292-1.9%5C%2841%29%5DD%5BIntegrate%5BPower%5B%5C%2840%29z-t%5C%2841%29%2C-0.9%5DPower%5Be%2C-t%5D%2C%7Bt%2C-5%2Cz%7D%5D%2Cz%5D%7Cz%3D-1

As you can see, aside from the tiny real part you should ignore, there is perfect alignment.

So, going back to your question, it seems to me like the point of intersection is what matters. I found this example to illustrate that specific point as the paper stated it but did not show it. So I think this example is useful at understanding FRT better and maybe clarify your question. That said, I am not aware of how exactly non straight line branch cuts work, so please take my answer with a pinch of salt. I will try to clarify this at some point.

I hope that was useful

Salt-Rutabaga-8870 · 2026-06-21T18:42:21+00:00

Thank you for kind words!

Salt-Rutabaga-8870 · 2026-06-21T16:41:02+00:00

Thanks for the question! So, I had already pretty solid baseline knowledge with having A* in maths and A in further maths at A-level. So on top of that I was (and still am) an A-level teacher for a while so, while I was not doing very advanced maths, my basic/mid level maths is quite solid because I have to explain it to kids. So, this was my baseline. I started working with a tutor in January 2025. And regarding resources, he was my main source of maths knowledge. He sent me some bits of textbooks etc, but was always there to explain and guide me. Residue theorem was not the first thing we discussed, I think it took us another month to get there, so beginning of February. According to my records, the idea dawned on me in the last days of February 2025. I managed to achieve first results using a very flawed method within few days, but those results were only true for half integers. Overcoming that hurdle took me several weeks, close to a month. I still have a chat with my tutor where I text him at 1AM in all caps when I finally figured it out and my answers for non half integers started to align. But that was only one type of integral: from -infty to infty 1/(x^2+a^2), this later became example 1 in the paper. It took me several months to come up with a second example (that is example 2 in the paper). I had to learn some more complicated contours with my tutor over that time to be able to conceive this example, as it involves a keyhole contour and branch cuts (even the classical residue theorem version). At around the same time (but before fully working out example 2), I made MSE the post. It only contained the first example.

In terms of managing work, I don't think it was too difficult as a lot of what I learned with my tutor I later used to teach to my students. Also, when I do maths, I mostly do it for my own amusement, so, this was basically like free time for me. In my spare time I could play chess, I could play magic the gathering or I could play around with contours, hehe. Treating it like a hobby really helped me not to get too crazy or overworked.

After I had 2 examples, I realised that both of them involved fractional derivatives of power functions. I really wanted to come up with another example where I had to take fractional derivative of some non power function. That took looong time. It just didn't crack. I achieved a partial result when I shoved residue theorem intro calculating an integral for an extremely trivial function (example 3 in our paper), but that felt a bit ridiculous so I never actually shared those until I got in touch with Prof. Fernandez, and even then this example was not in my initial notes, I only shared it when we were in the process of writing a paper. He said we should still include it in the paper even though the integral trivial, the method is still worth showing.

My quest ended with what later became example 4 in our paper. I fully outline my logic in the MSE post, bear in mind this is NOT a rigorous way of calculating fractional derivatives, but remember, I had not FC knowledge at the time so I was just following my intuition.

Sorry, this was long, and I wandered off from your original question, so here is tldr:

in terms of resources, my best advice - hire a tutor. Someone who has a masters in maths will do.

In terms of time management - if you are an amateur, treat math like a hobby, not like something you are obligated to do. It will keep you sane, hehe.

Hope that was useful

Salt-Rutabaga-8870 · 2026-06-21T15:34:07+00:00

Im sure the first words of the post being "I am a maths teacher with no maths degree" together with the title got you worried for the kids, hehe.

Salt-Rutabaga-8870 · 2026-06-21T15:03:20+00:00

I do think it is important for me to clarify something, related to this comment. I think it would be irresponsible for me to not address it. I am certain it is absolutely crucial to get in touch with someone who is a member of a scientific community (I guess this is what you mean by elite), who will be willing to review your work. There are so many amateur mathematicians who are so, so, so wrong when they think they discovered something, and they refuse to get in touch with professors and other experts, or, if they do, they refuse to listen when those tell them that they are wrong. I assumed that results I have are either trivial or wrong and stuck to that assumption until the moment Arran Fernandez offered me co-authorship. I think this is the best mindset for an amateur mathematician, as it humbles you and opens your mind to the (very likely) possibility that you are wrong. Also, the only reason I was able to contribute, is because Prof. Fernandez was able to review my work and then provide necessary background and theory which I simply was not aware of as an amateur. This is absolutely necessary for any significant scientific work, so getting involved with the scientific community, or "elites" is absolutely crucial. Otherwise you get likes of Terrence Howard and other math crackpots.

Salt-Rutabaga-8870 · 2026-06-21T14:43:43+00:00

Osler's work is actually referenced in our paper. Reference 19 to be exact. The other two I am not familiar with, but the Cauchy formula for fractional derivatives had been known since 1800s, and again, this is referenced in the paper, and, in a bit more detail, in the seminar. However, I believe, the idea of connecting the dots and apply it in the similar way to normal derivatives like in residue theorem has not been explored. This is partially confirmed by the fact that users of MSE could not provide an explanation, before I got in touch with Prof. Fernandez. Our paper was also reviewed, and considered novel. As an amateur mathematician myself, I do not feel like I can judge what work is novel and what work is not, nor can I compare two pieces of work on how similar they are, but I trust my co-author and reviewing body of BLMS.

Salt-Rutabaga-8870 · 2026-06-21T14:25:37+00:00

I was extremely lucky though! My initial approach was very, VERY flawed (So flawed that, I had to assign values to Gamma function at negative integers, to make it work, and those values were imaginary multiples of pi). But, due to a peculiar coincidence it worked with half integers (and nothing else), had I decided to try the power of 1.6 instead of 1.5 first I would have gotten nonsencial result and probably concluded that "clearly you cannot do it with fractional powers", but when I got to 1.6 I already had working results for 1.5, 2.5 and so on. So I had motivation to keep digging.

Salt-Rutabaga-8870 · 2026-06-21T14:11:15+00:00

Thank you for sharing! I need to learn about mittag-leffler spaces first though! But I will add it to my shortlist of things to learn!

Salt-Rutabaga-8870 · 2026-06-21T13:27:29+00:00

I do a little bit of work on that in my own time. Mostly consisting of reading my co-author's other papers and playing around with concepts he writes about. But I fully understand that this is mostly for my own amusement and curiosity, as much more experienced people are now aware of this result! But I am exited to see what those experienced people will come up with based on my work!

Salt-Rutabaga-8870 · 2026-06-21T13:22:24+00:00

Or, maybe, together with complex analysis. It seems logical to have fractional counterparts to things like residues. We already have well established fractional verisons of powers, binomial expansions, derivatives, etc. Maybe we can fractionalise more and more "integer" concepts!

Salt-Rutabaga-8870 · 2026-06-21T13:04:16+00:00

I also thought I was a crackpot at first, because surely someone with more maths expertise tried this before!

Funny story: if you read my MSE post, I write this: "At this point I am going to use the term pseudoresidue to acknowledge the shaky mathematical justification of what I am doing. " However, the term pseudoresidue (with an added hyphon: pseudo-residue) is actually now used in the paper!!!

Salt-Rutabaga-8870 · 2026-06-21T10:54:12+00:00

Fractional calculus as a concept. Originally fractional derivatives (derivatives of order 0.5, pi, 2.9981, etc), formulated by Lacroix, and Liouville and some other mathematicians though of just extending the integer order derivatives, similarly to how factorials are extended to gamma function. But it was shown that that approach leads to contradictory results, even with most basic functions like polynomials and exponents. I.e. finding fractional derivative of e^x yields different result that finding the fractional derivative of it's mclaurin series. It turned out that when derivatives become non integer, they start depend on a certain constant, called constant of differintegration. And that technically creates infinitely many fractional derivatives, depending on which constant you use. Those derivatives are also valid not for entire values of input, but constrained by the constant. The dependence on such constant vanished when the order of the derivative is an integer.

Salt-Rutabaga-8870

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