AofA
Analysis of the Computational complexity of Algorithms, i.e. find out how much time, and how much memory does an algorithm take to run.
Worst case analysis of algorithms
Average case analysis of algorithms
Amortized analysis of algorithms
Sometimes asymptotics do not give the practical complexity
Analytic Combinatorics, Part I (Analysis of Algorithms)
Already recognized as important by Babbage, Turing. However, the modern field of analysis of algorithms was started by Donald Knuth, who recognized that mathematics had the tools to analyze algorithms. Things like the following are useful tools for this:
Books: four volumes of The art of computer programming.
See examples in: Towers of Henoi, Binary search