Welcome to Week 3 of "Analytic Combinatorics."
This week, we introduce the idea of viewing generating functions as
analytic objects, which leads us to asymptotic estimates of
coefficients. The approach is most fruitful when we consider GFs as
complex functions, so we introduce and apply basic concepts in complex
analysis. We start from basic principles, so prior knowledge of
complex analysis is not required.
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Lecture 4: Complex Analysis, Rational and Meromorphic Asymptotics. We
consider basic principles of complex analysis, including analytic
functions (which can be expanded as power series in a region);
singularities (points where functions cease to be analytic); rational
functions (the ratio of two polynomials) and meromorphic functions
(the ratio of two analytic functions). The heart of the matter is
complex integration and Cauchy's theorem, which relates coefficients
in a function's expansion to its behavior near singularities. The
discussion culminates in a general transfer theorem that gives
asymptotic values of coefficients for meromorphic (and rational)
functions.
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There are two clusters of errata in this lecture. See ACqa3.pdf for details.
Your assignment for this week, due at 11:59PM on
Thursday, April 7, 2022
is to write up and submit solutions to
Web Exercise IV.2 and Note IV.28 on the "Analytic
Combinatorics" booksite. For extra credit, do Program IV.2.
If you're more comfortable with some other package for plotting complex
functions, feel free to use it.
As usual, submit a potential exam question on the week's material.
Submit files named
"AC3-Q1.pdf"
"AC3-Q2.pdf"
"AC3-Q3.pdf" (optional extra credit)
"AC3-QQ.pdf"
via codePost.
RS