questions on nonelementary antiderivatives and Risch algorithm

Question

I'm not sure if my mathematical knowledge allows to understand things correctly, so please correct me if I'm wrong.

The existence of an algorithm, which decides whether the given function has an elementary antiderivative or not, may exist or not - depending on what "elementary function" means?

Because we have https://en.wikipedia.org/wiki/Risch_algorithm however "if one adds the absolute value function to the list of elementary functions, it is known that no such algorithm exists; see Richardson's theorem" ("Decidability" in the wiki article cited above)?

If that is true, in what meaning of "elementary function" does the Risch algorithm work?

https://en.wikipedia.org/wiki/Elementary_function With solutions of algebraic equations, in particulary roots, included, I thought the function $|x|=(x^2)^{\frac{1}{2}}$ is elementary?

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One more question:

http://mathworld.wolfram.com/LiouvillesPrinciple.html I'm not familiar with the differential field theory. What does the Principle say in standard case, I mean for continuous real functions defined on intervals? Is this some kind of generalisation of the famous Liouville's theorem?

Answer 1

You are right, "elementary function" must be defined properly. For example, the definition could something beginning like this:

A holomorphic function $f(z)$ defined on a domain in $\mathbb C$ is called an elementary function if ...

Thus, in particular, things like $|x|$ in the real line or even worse the function $$ f(x) = \begin{cases}1, & x\text{ is rational}\\ -1, &x\text{ is irrational}\end{cases} $$ are not included. Even though $f$ satisfies the polynomial equation $f(x)^2=1$, it is not considered an "algebraic function".

One should beware of "amateur" definitions, such as the one on Wikipedia!

Answer 2

Unfortunately, I cannot provide a satisfactory answer to the question : What "elementary function" means?

The notion of "elementary function" seems rather conventional. Also conventional, the "standard" special functions. In fact, to avoid ambiguity when we deal with elementary or special functions, we should refer to lists of functions respectively written down.

Some thoughts about this subject are presented on a simplified overview in the paper (English translation pp.18-36) https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales

As far as I know, the original Risch algorithm was dealing with algebraic, trigonometric, exponential, logarithmic functions. The actual extensions are not limited to elementary functions only. Nowadays, it is difficult (quite impossible for the commun user) to know exactly on what list of functions a symbolic integration software deals with.

WolframAlpha includes in the list of basic functions $abs(x)$ and $sgn(x)$ and many other of course. For example the answer to "Integrate abs(x-2)exp(x)" is $e^2-(e^x(x-3)+e^2)\text{sgn}(2-x)$ where sgn means sign of.