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?

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

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

Rosenlicht, M. (1972). Integration in finite terms. American Mathematical Monthly, Vol. 79, No. 9 (Nov., 1972), pp. 963-972. JSTOR, Google

Liouville's Theorem is used at the end of the paper to show precisely the fact about which you have asked. Here is an excerpt:

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