I think, there's no (preferrably real) analytical solution to the equation
$$x+e^{-x}=2,$$
however I have no idea how I'd proof that guess. How does one generally proof the nonexistence of an analytical solution?
A Closed-form expression is defined by Wikipedia as
a mathematical expression expressed using a finite number of standard operations.
I assume an analytical expression doesn't necessarily have to include only a finite number of operations – i.e. an infinite series $\sum_{K=0}^\infty a_k$ is also an analytical solution, if it is convergent? I didn't find much information on a precise definition of analytical expressions.
If that definition is in fact correct, one could argue though that there is an analytical solution. Because the equation is clearly numerically solvable, hence there exists a sequence $x_k$ that approximates the solution $x$, so $x$ can be written as
$$x=x_0+\sum_{k=1}^\infty(x_k-x_{k-1})$$
So maybe my question should be how to proof that there is not Closed-form expression that solves above equation.
This seems similar to the Abel-Ruffini theorem, which states that no algebraic solution (a subset of the closed form expressions if I understand it correctly) to polynomials of degree 5 or higher.
However, I don't want to limit the allowed solutions too much. Simply said I want to proof that you can't solve the equation exactly with just a pen and paper and a finite number of transformation steps, like subracting $x$ from both sides.
Best Answer
There are two real solutions:
$$x_1=W\left(-\frac{1}{e^2}\right)+2,$$
$$x_2=W_{-1}\left(-\frac{1}{e^2}\right)+2,$$
where $W(x)$ is the Lambert W-function.