>>11278664Let me back up: to prove a number is transcendental, generally the procedure is to look at the series expansion and show that it cannot be represented the root of a (finite) algebraic equation. However, most numbers on the real line are transcendental, and for any arbitrary number, coming up with a proof is usually more difficult than for pi or e.
Similarly, to show that an equation is transcendental, you show that it cannot also be represented as a finite algebraic equation. Our old favorite functions sin(x) and e^x are both transcendental, but they are considered special cases because they are well behaved and commonly arise (also, they are solutions to first and second order ordinary (algebraic) differential equations, but that is quite irrelevant here). Even though e^x and sin(x) are well behaved, because of their non-algebraic nature, we must also define new functions for their inverses, y=ln(x) and y=arcsin(x).
If you're looking for the closed form (or the inverse of the closed form) of a given transcendental equation in terms of commonly-known functions, it is generally impossible ipso facto. For the specific case of the inverse of the Lambert W, it is because we are not able to separate the x and e^x terms. Take note as well that it has no inverse because it is not one-to-one.
