Using classical hypergeometric identities, we compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. We deduce several explicit computations on the related Green function and Martin kernel. In the second part of the talk, I will consider the two-dimensional Markov process constructed on the stable Lévy process and its area process, and give an explicit formula for the harmonic measure of the split complex plane. This formula allows to compute the persistence exponent of the area process and to solve a problem raised by Z. Shi. If time permits, I will display a possible connection between this persistence exponent and the Hausdorff dimension of the Lagrangian points of the inviscid Burgers equation with Lévy stable initial data. This is based on two joint works with Christophe Profeta (Evry).