6.   Singularity Analysis of Generating Functions

VI.1 A glimpse of basic singularity analysis theory

VI.2 Coefficient asymptotics for the standard scale

VI.3 Transfers

VI.4 The process of singularity analysis

VI.5 Multiple singularities

VI.6 Intermezzo: functions amenable to singularity analysis

VI.7 Inverse functions

VI.8 Polylogarithms

VI.9 Functional composition

VI.10 Closure properties

VI.11 Tauberian theory and Darboux’s method

VI.12 Perspective

Web Exercises


Use the standard function scale to directly derive an asymptotic expression for the number of strings
in this following CFG: S = E + U×Z×S + D×Z×S, U = Z + U×U×Z, D = Z + D×D×Z.


Give an asymptotic expression for the number of rooted ordered trees for which every node has 0, 2, or 3 children. How many bits are necessary to represent such a tree?


(D. Carter) A rainbow tree is a rooted ordered tree where each node is one of four colors, with the rules that each node differs in color from its parent and can only have children which all differ in color from each other. (Note that no node can have more than 3 children.) Give a ~-expression for the number of rainbow trees with $N$ nodes.

Selected Experiments

Program VI.1

In the style of the plots in Lecture 6, do $r$- and $\theta$-plots of $1/\Gamma(z)$ in the unit square
 of size 10 centered at the origin. Click here for access to the Java code used for the lecture.