Prove $f(x)=\int\frac{e^x}{x}\mathrm dx$ is not an elementary function

Question

How do I prove that the exponential integral $$f(x)=\int \frac{e^x}{x}\mathrm dx$$ is not an elementary function?

Also, what are the general methods and tricks to prove that an integral or solution to an equation is not an elementary function?

Answer 1

All we need for this is a theorem of Liouville (1835): Suppose that $f$ and $g$ are rational functions with $f\neq 0$ and $g$ non-constant. Then $$\int f(x) e^{g(x)} dx $$ is an elementary function if and only if there exists a rational function $r$ such that $ f=r'+g'r.$

Here we have $g(x)=x$ and $f(x) = 1/x.$ Assume there exists a rational function $r$ such that $ 1/x = r' + r \text{ } $ (1). Denote the multiplicity of the pole at $0$ by $m$ (where $m\geq 1$ so that both sides of (1) agree when $x\to 0$), so that $ \displaystyle r(x) = \frac{p(x)}{x^m Q(x)} $ where $p, Q$ are polynomials with no common factors and $Q$ is not divisible by $x.$

Substituting this form into (1) and multiplying both sides by $x^m$ yields $$ x^{m-1} = \frac{p(x)+ p'(x)}{Q(x)} - \frac{Q'(x) p(x) }{Q^2(x)} - \frac{m}{x Q(x)} .$$ Taking limits of both sides as $x\to 0$ illustrates the lunacy in our assumption that such a rational function exists, and hence $\displaystyle \int \frac{e^x}{x} dx$ is non-elementary.