# Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful tool for evaluating sums, essentially a systematization of the use of telescoping sums. Why isn't it more widely known and used? (Related question)

• I think it has no version of the chain rule. Which is a pretty big disadvantage, to me. – Akiva Weinberger Feb 10 '16 at 3:43
• @AkivaWeinberger false, it does have a very obscure chain rule, but it is significantly trickier to use – frogeyedpeas Feb 10 '16 at 3:49
• Let $D_{h,x}(f) = \frac{f(x+h)-f(x)}{h}$ then $D_{h,x}(f(g)) = D_{h,x}(g) D_{h*D_{h,x}(g),g}(f(g))$ – frogeyedpeas Feb 10 '16 at 3:50
• The challenge here is that unless $h=0$ or some function that is known to cancel out with the difference g, then you end up having to recursively work with sub-difference equations, in order to say make a change variables (like a u subsitution) etc... that being said it is doable with enough machinery built up – frogeyedpeas Feb 10 '16 at 3:52
• It's thriving as q-calculus or quantum-calculus. See mathematicalgarden.wordpress.com/2008/12/15/what-is-q-calculus, en.wikipedia.org/wiki/Quantum_calculus, and surveys by Ernst on the topic. Academia is incredibly conservative and much too rigid to introduce anything but thoroughly mainstream topics to students (yawn). See also the finite operator calculus (Rota, Roman, ...) and umbral calculus, as well. – Tom Copeland Feb 10 '16 at 4:27

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources.

The name "finite calculus" is unusual. The traditional term is calculus of finite differences or variants such as difference calculus. A search for those terms will be more productive.

It seems to me an incredibly powerful tool for evaluating sums, essentially a systematization of the use of telescoping sums. Why isn't it more widely known and used?

It is widely known and used, most obviously in numerical analysis, but also in many other subjects.

• Thank you for ending my search term scavenger hunt! I was already suspecting it had a different name. – Melvin Roest Apr 25 '19 at 18:07

All formulas of discrete calculus will depend on $\Delta x$ (and $\Delta y,\Delta z$, etc.). After $\Delta x\to 0$, the formulas will inevitably become simpler. Those more cumbersome formulas may be the main turn-off, especially for the people who already know calculus well. On the other hand, the limits themselves seem to be an even bigger turn-off, especially for the people who don't know calculus yet.

Using finite differences involves a considerable complication of the procedures of the calculus. Such an approach may be constructively more satisfactory but may not be appropriate for a first acquaintance with the subject. The technical complications of the epsilon-delta approach are involved enough, and it is how certain things are defined (and it is way easier to do things rigorously with) and furthermore it is the received way of doing things.

• On the other side, finite calculus doesn't have to deal with technical complications as Jordan curve theorem etc... :) – Lehs Feb 18 '16 at 8:31
• @Lehs, actually Kanovei has a combinatorial proof of this using infinitesimals. – Mikhail Katz Feb 18 '16 at 14:28
• That sounds interesting, but I doubt I would understand it... – Lehs Feb 18 '16 at 15:07