Here are some hints which might be useful. Far from being an expert in this field, I by myself depend strongly on accessible information. Fortunately there are some good books from the great providing appropriate information.
A first step:
Mathematics for the Analysis of Algorithms from D.E.Knuth and D.H.Greene
This is a small booklet providing you with a nice survey on interesting techniques and examples of algorithms and their mathematical analysis. Chapter 4, Asymptotic Analysis is a good starter on this subject also showing some instructive examples in about 35 pages.
The next one is much more comprehensive.
Analytic Combinatorics is a classic from P. Flajolet and R. Sedgewick.
This book consists of two parts. The first part A treats formal power series which is not of central interest for you. But the second part B: Complex Asymptotics provides you with a wonderful presentation of analytic methods on power series on more than 500 pages starting with a refresher in complex analysis (analytic functions).
I deeply appreciate this book for the presentation of the material, the wealth of examples and last but not least for a treasure of many hundreds of valuable references together with interesting information on them.
Note: You may get an appropriate impression of the content when looking at a (draft) version of some of the chapters. Please note, that the final book version contains even more material.
The references in the book often point to more sophisticated material. One of these references (ref. ) points to a Survey on Mellin Transforms which might also be of interest for you.
Another reference in this book, namely ref.  (and also referred to by D.E.Knuth and D.E.Greene above) points to a classic of P. Henrici which is my next hint for you.
Applied and Computational Complex Analysis, vol.2 is a true classic on Applied Complex Analysis from 1977. Chapter 11: Asymptotic Methods provides accessible information and is a pleasure to study. Especially section 11.8: The Method of Steepest Descent was very helpful for me for a better understanding of the saddle point method.