For example: Lambert $W$ is a non-elementary function that can be defined as a solution for $x$ to $x\cdot e^x$, but $\int{\frac{1}{\ln(t)}dt}$ is also supposed to be nonelementary. How do we know that those are not related by known operations? When is it necessary to define a new second or third etc. function?
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1$\begingroup$ Not sure whether we can decide this in general. Every time we add a new function we increase the set of elementary functions in the sense that the new functions are allowed as well. Maybe, some tree-structure like in the case of the Risch-algorithm works. $\endgroup$– PeterCommented Feb 25, 2022 at 10:49
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$\begingroup$ @Peter do you know how could I tag it? or for what to look for? If it is an unsolved problem then examples would be enough for me. $\endgroup$– kontoCommented Feb 25, 2022 at 11:29
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1$\begingroup$ "Supposed" is a wrong word to use. The said integral is known to be non-elementary. That's also the answer to your final question. $\endgroup$– Ivan NeretinCommented Feb 25, 2022 at 13:31
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$\begingroup$ Actually the product log is elementary according to Wolfram functions. Scroll to the bottom of the page where it says “product logarithms”. Also, we do not need functions since we could use series expansions of function which would be equivalent. $\endgroup$– Тyma GaidashCommented Feb 25, 2022 at 13:55
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$\begingroup$ @TymaGaidash maybe Wolfram functions has a mistake? tandfonline.com/doi/abs/10.1080/10652460802332342 (assuming this paper is correct, I didn't look at the proof) $\endgroup$– JojoCommented Feb 26, 2022 at 7:23
2 Answers
I edited your question from "How do we know that those are not related?" to "How do we know that those are not related by known operations?", because two functions are always in a relation.
Liouville's theorem (in differential algebra) together with Risch algorithm helps to decide if the antiderivative of a given elementary function is an elementary function. In [Risch 1979], a structure theorem for algebraic independence (contains non-identity) of elementary functions and an associated algorithm is proven. Liouville's theorem is a special case of that structure theorem.
At least for some other classes of functions that contain the Elementary functions and some Special functions, comparable structure theorems are available.
Starting with an elementary function, repeated antiderivation generates the set of Liouvillian functions. [Risch 1979] brings a comparable structure theorem of Leo Königsberger for the Liouvillian functions. [Rosenlicht 1969] proves a decision theorem for inverses in the Liouvillian functions.
The Elementary functions are a proper subset of the Liouvillian functions. The Elementary functions are closed under algebraic operations, composition and differentiation. The Liouvillian functions are closed under algebraic operations, composition, differentiation and indefinite integration.
Your example $\int\frac{1}{\ln(t)}dt$ yields a non-elementary Liouvillian function.
[Bronstein et al. 2008] proves that Lambert W is not a Liouvillian function. The $n$-th derivatives and the $n$-th antiderivatives of Lambert W are rational functions of Lambert W and the identity function. Because Lambert W is not a Liouvillian function and the Liouvillian functions are closed under algebraic operations,the $n$-th derivatives and the $n$-th antiderivatives of Lambert W are not Liouvillian functions therefore.
Only a few mathematical problems regarding the belonging of considered functions to known classes of functions and the relations between classes of functions are solved. The reason is that only a few theorems and conjectures about algebraic independence of numbers are known in transcendental number theory (e.g. Schanuel's conjecture).
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Here is a theorem about a necessary condition for a function having an elementary antiderivative:
https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)
Here is a semi-algorithm (not a full algorithm) used to compute the antiderivative of functions:
https://en.wikipedia.org/wiki/Risch_algorithm
Mathematicians can study a particular function and give it a new name, without having to check whether or not it is an elementary function. The fact that the Lambert-W function or the gamma function is not elementary can be curious in some respects. But those studying e.g. complex analysis don't need to know that for their work.
I'll give you an analogy that might shed light on this. It is not known whether the following constant is transcendental or not (though amazingly it has been proven to be irrational):
https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant
That wouldn't stop e.g. a number theorist from making use of this constant.
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2$\begingroup$ I don't think this answers the question. I read the question as 'is it possible to know when two different special functions are independent, up to relations involving elementary functions', with $\int dt/\ln(t)$ just given as an example of a special function $\endgroup$– JojoCommented Feb 25, 2022 at 19:54
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$\begingroup$ @Joe, yes you put it well, thats what I ment! $\endgroup$– kontoCommented Feb 25, 2022 at 21:00