Complex analysis has long been regarded both as an elegant subject in its own right, and as a powerful tool in applied mathematics. The subject has inescapable limitations, however, and it might be said that it has been superseded by more modern methods. I aim to show here that, even taking these limitations into account, there is a niche for it in modern mathematics, in this case as applied to industrial problems. The modernity lies perhaps more in the novelty of the problems studied (here, for example, free boundary problems from industry) and in the interaction with, say, numerical methods than in the techniques themselves.