# Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$.

Now using this fact: How should I proceed to prove that $\int \dfrac{\sin z}{z}dz$ cannot be written using elementary functions?

• Hint: try multiplying by a clever form of $1$ ($e^{-z}e^z$). – Cameron Williams Mar 1 '14 at 1:50

To see why the antiderivative of $\sin(z)/z$ cannot be written using elementary functions, consult: