Why Lévy α-stable distributions lack general closed-form expressions for arbitrary α

The ubiquitous Lévy α-stable distributions lack general closed-form expressions in terms of elementary functions-Gaussian and Cauchy cases being notable exceptions. To better understand this 80-year-old conundrum, we study the complex analytic continuation p_{α}(z), z∈C, of the symmetric Lévy α-stable distribution family p_{α}(x), x∈R, parametrized by 0<α≤2. We first extend known but obscure results, and give a new proof that p_{α}(z) is holomorphic on the entire complex plane for 1<α≤2, whereas p_{α}(z) is not even meromorphic on C for 0<α<1. Next, we unveil the complete complex analytic structure of p_{α}(z) using domain coloring. Finally, motivated by these insights, we argue that there cannot be closed-form expressions in terms of elementary functions for p_{α}(x) for general α.