Mathematical analysis and computational complexity?
Flajolet and Sedgewick published the book "Analytic Combinatorics" http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html.  I don't know much about that topic, but people in the field use tools from complex analysis.  So far, their applications seem more on analysis of algorithms, not on computational complexity, as far as I see.