TITLE: Why can't erf be expressed in terms of elementary functions?
QUESTION [5 upvotes]: I have seen this claim on Wikipedia and other places. Which branch of mathematics does this result come from?

REPLY [6 votes]: This is a consequence of the Liouville-Ritt theory of integration in finite terms. You can find a brief sketch and references in my post here.

REPLY [4 votes]: I am not aware of any easy proof of this. Here is a proof that $\int e^{x^2}$ cannot be expressed in terms of elementary functions:
http://math.hunter.cuny.edu/ksda/papers/rick_09_02.pdf

REPLY [3 votes]: Differential Galois theory.