When is it necessary to define a new function?

Question

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?

Answer 1

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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[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Rosenlicht 1969] Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22

[Bronstein et al. 2008] Bronstein, M.; Corless, R. M.; Davenport, J. H.; Jeffrey, D. J.: Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville. Integral Transforms and Special Functions 19 (2008) (10) 709-712

Answer 2

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.