# Errata for An Invitation to Analytic Combinatorics

Below is the known errata for An Invitation to Analytic Combinatorics: From One to Several Variables by Stephen Melczer – see the textbook website for the textbook manuscript and associated computer algebra worksheets.

Please email any errors you notice to the author.

• Due to editor error the original version of the text posted to the publisher website had an incorrect title and author affiliation. This has been fixed as of February 2021 and the correct title should appear in all printed copies.

• In Section 1.1.2 of the Introduction, the second step set $\mathcal{S} = \{(0,-1), (\pm1,1) \}$ should be $\mathcal{S} = \{(0,1), (\pm1,-1) \}$ in order for $b_n$ to have the stated asymptotic behaviour.

• Corollary 2.1: The exponents on $\beta$ and $\omega_j$ in the displayed asymptotic formula should be $-n$ instead of $n$. The correct formula is $$f_n = \frac{\beta^{\color{red}{-n}}n^s}{\Gamma(s+1)} \sum_{j=0}^{m-1}C_j\omega_j^{\color{red}{-n}} + O(\beta^{\color{red}{-n}}n^t)$$

• Proposition 2.19: The statement that the indicial polynomial has only rational roots should be read to mean that the indicial polynomial at any singularity has only rational roots.

• Example 3.12: The final term in the polynomial $P$ should be negated. Corrected, the polynomial is $P = z − xy − (x + y + xy)z \, {\color{red} - xz^2}$.

• Definition 3.9: The inequalities in the final sentence should all be strict. In other words, the final sentence should read: Equivalently, a minimal point $\mathbf{w}$ is a singularity of $F$ such that no other singularity $\mathbf{z}$ of $F$ satisfies $|z_j| < |w_j|$ for each $1 \leq j \leq d$.

• Example 5.6: The asymptotic expansion at $\sigma$ has errors in its first and fourth terms. Corrected, the asymptotic contribution of $\sigma$ is $$\Phi_{\sigma} = 4^n \left( \frac{{\color{red}4}}{\pi n} - \frac{6}{\pi n^2}+\frac{19}{2\pi n^3} - \frac{{\color{red}63}}{4\pi n^4} + O\left(\frac{1}{n^5}\right)\right).$$ The final asymptotic expansion of $f_{n,n,n}$ in the text is correct as presented.

• Example 5.7: There is an extra factor of $s$ in the denominator of the leading constant. The leading constant should read $(rn)!(sn)!/\sqrt{2 \pi ae^{-a}(be^{-b}+ae^{-a}-ab)}$