In Chapters 39 and 40 we shall see how Sturm and Liouville extended the ideas of Fourier discussed in the last chapter. Liouville worked in many other mathematical fields as well and his results here were often so simple and basic that they have been completely absorbed into the general body of mathematics. It thus seems appropriate to recall some of his achievements.

At the age of 27 he founded the Journal des mathématiques pures et appliquées, which became, under his editorship, one of the major journals of the nineteenth century. He edited and published Galois' manuscripts in his journal and, just as importantly, gave a series of lectures organising and interpreting Galois theory for the general mathematical public.

In complex variable theory he used simple general arguments to bring order to the subject of multiply periodic functions. The result, called by his name which states that a bounded analytic function is constant, was known to Cauchy, but Liouville was the first to demonstrate its power. Some of the flavour of his work in this field is conveyed in the following theorem.