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In this paper, we propose a class of irregular turbo codes that approach the capacity of the binary erasure channel. First, an analytic expression of the erasure probability of punctured recursive systematic convolutional codes is derived. This expression will then be used to study the density evolution of turbo codes over the binary erasure channel, that will allow for the design of capacity-approaching infinite-length irregular turbo codes. Next, a graph-optimal interleaver for finite-length irregular turbo codes is proposed. Finally, simulation results for different coding rates are shown.