Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their popularity plummeted. I could hardly find anything on the net on Abelian functions and Wikipedia thinks that they are barely worth mentioning. What did the 19th century theory of these functions consist of, and why was it so important?



Why were Abelian functions so important in the 19th century?
For a really detailed answer to your question, see *The Legacy of Niels Henrik
Abel*, edited by O.A. Laudal and R. Piene (Springer 2004). In particular, there is a long introductory article by Christian Houzel, most of which can be viewed [here](http://books.google.com.au/books?id=HiXwhBm42hcC&lpg=PA21&ots=ibJuErm5sc&dq=%22christian%20houzel%22%20abel&pg=PA21#v=onepage&q=%22christian%20houzel%22%20abel&f=false).


**Addendum.** The complete article "The Work of Niels Henrik Abel" by
Christian Houzel may be found [here](http://www.abelprisen.no/nedlastning/litteratur/houzel_the_work.pdf).


