Basic Mathematics - 2020-12-13

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Billy Rubina

06:31

@user21820 It says that: "there is an algorithm that will numerically evaluate the integral of any computable function." But isn't that different from something like finding an antiderivative? Or they are the same thing? When I did numerical analysis, there are functions where you can't find antiderivatives but you can use numerical methods to approximate it, If I remember well.

user21820

@BillyRubina You're getting your concepts wrong here. An antiderivative of f is defined as a function F such that F' = f. Nothing more or less than that. And nothing to do with whether you can compute it or not.

Curiously, there are differentiable functions whose derivative cannot be Riemann-integrated. But every continuous function f has an antiderivative whose derivative is f, by FTC.

> there is an algorithm that will numerically evaluate the integral of any computable function

To explain this quote you firstly need to know what "computable function" means, otherwise it is vague or meaningless. Let CR be the set of computable reals, where a computable real is a program that on input k∈ℕ outputs the integer part if k=0 and outputs the k-th binary digit if k>0. This is just one possible definition of CR; many others are equivalent, such as via computable Cauchy sequences or computable regular Cauchy sequences.

A computable function f in the context of real analysis is then a function from CR to CR that is itself computable; f is a program that on input p∈CR outputs the program q∈CR that represents the value of the function on p.

Note that any computable function is necessarily continuous, otherwise you can computably determine whether a computable real is positive or not, which is impossible.

The claim on wikipedia then is that there is a computable function I such that for every (continuous) computable function f : CR→CR we have that I(f) is differentiable and its derivative is f.

The reason I thought it was impossible is that with any finite sampling there is no way to figure out whether there is some sharp spike somewhere, so how can it be possible for such an I to exist?

So either wikipedia is wrong, or they are using a sufficiently restricted notion of "computable" or "continuous" (for which I already linked you to a notion of controlled continuity that would suffice), or there is something wrong with my intuition. I don't think the last possibility is the case.

Billy Rubina

07:21

@user21820 Perhaps, what I am talking about is "an antiderivative in terms of elementary functions", this doesn't always exist, right?

user21820

@BillyRubina Yes. ∫ exp(−x^2/2) dx is a famous example. I was just pointing out that "anti-derivative" really has nothing to do with computability or approximability or anything to do with how easy humans can get it.

And note that elementary functions are only a small subset of computable functions. It's a bit arbitrary what functions are considered elementary, but so far people decided to include all functions that can be generated from the complex exponential function and polynomial functions by composition and inverse.

Liouvillian function

In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, it is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives. The logarithm function does not need to be explicitly included since it is the integral of 1...

Billy Rubina

07:44

@user21820 I guess I mixed things up and thought that "antiderivative" already means "antiderivative in terms of elementary functions".

user21820

Yea it's common because at high-school level one is taught various antiderivatives to solve certain basic integrals, but only those which can actually be solved by elementary functions are given! Nobody ever tells us clearly that there are functions that do have antiderivatives that are not elementary. So no sample data to learn that. =)

Billy Rubina

High-school? Where are you from?

user21820

@BillyRubina Sorry, I don't reveal where I am from online. =)

Billy Rubina

Correction: In Brazil, seeing integrals and derivatives on HS is "science fiction".

@user21820 We don't see it at all.

user21820

@BillyRubina Oh I see.

Billy Rubina

07:50

I mean, we see a bit of that on physics, but it's never mentioned they are derivatives or integrals.

user21820

That's interesting. So there is no calculus at all?

Billy Rubina

Yeah.

I don't know if things have changed, but I think this didn't change. I assisted some HS students in the past years and never seen they mentioning integrals or derivatives.

user21820

I see.

I don't have much liking for calculus at high-school level, because of the lack of rigour, but if it is done rigorously I think it is useful. For me the issue is simply the matter of when the students are ready for rigorous calculus.

Billy Rubina

I am looking at my old physics textbook from high-school: Formulas in mechanics can always be derived from a small set of formulas, right? For example: We have $F=ma(t)$, by integrating we get velocity and position, right?

And it's kinda easy to integrate simple examples of polynomial functions, which is all that is seen on HS here.

In the book, they give 3 formulas.

If in physics, we had a function that when integrated repeatedly, we could obtain other 175 functions which are physically relevant, I guess they'd probably make the students memorize 175 functions.

user21820

08:31

@BillyRubina Haha yea.

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