Analytic Combinatorics/Meromorphic Functions Residue Method

Introduction Armed with our new understanding of how we might use contour integration to find the estimates we want, we can give a new proof of our results for meromorphic functions. Theorem We restate the theorem from that chapter. If is a meromorphic function with pole singularities of orders respectively where has a modulus strictly smaller than all the others, then Proof By Cauchy's coefficient formula, if is a small circle around the origin that does not contain a singularity of , then . 400px This is what we want to estimate. Draw another circle centered at the origin but of a larger radius , big enough to encompass all singularities in its interior. By Cauchy's inequality, which can be made arbitrarily small by making arbitrarily big. 400px For each singularity, draw small circles centered at each . 400px We construct a single contour of integration out of our circles as such: starting from some point on and travelling anti-clockwise for each and travelling along a straight-line which joins to our small circle travelling clockwise all the way around our small circle until we get back to where we first reached it travelling in the opposite direction along the same straight-line we got here We do this until we ending up back where we started at the very beginning. 400px This is a closed contour whose interior contains no singularities. Therefore, Cauchy's theorem tells us that Travelling in one direction along the lines and back again cancels out (see the final property of contour integration). So, removing them from the above equation and rearranging gives . We can calculate using the technique of residues. Using the Laurent expansion of , we can write the integral around any of our circles . For any , does not have a singularity and, by Cauchy's theorem again, . For negative values of , set and switch to polar coordinates, , and . For any contour enclosing , for .Stroud 2003, pp. 883. For because and for all integers .Stroud 2003, pp. 884. Therefore, because all the terms in the above sum are zero except when , .Stroud 2003, pp. 923. is called the residue of our function at , which we denote We can calculate the residue .Stroud 2003, pp. 925. Because is a pole of order then and so the Taylor expansion of at can be written . Therefore, . To differentiate, we apply the product rule The largest term in the sum on the right-hand side is Taking gives us . We calculate the residue of every pole to finally get our stated resultMishna 2020, pp. 96-97.Pemantle, Wilson and Melczer 2024, pp. 81-82.Flajolet and Sedgewick 2009, pp. 259. Notes References Flajolet Philippe Sedgewick Robert Analytic Combinatorics Cambridge University Press 2009 https://ac.cs.princeton.edu/home/AC.pdf Mishna Marni Analytic Combinatorics: A Multidimensional Approach Taylor & Francis Group, LLC 2020 Pemantle Robin Wilson Mark C. Melczer Stephen Analytic Combinatorics in Several Variables Cambridge University Press 2024 2nd https://acsvproject.com/PemantleWilsonMelczer23.pdf Stroud K. A. Advanced Engineering Mathematics 4th Palgrave Macmillan 2003