Logarithmic terms in series expansions

Question

Lately, I am encountering more and more example of series expansions that do include logarithmic terms.

For example (section 1 (page 3-5) in hep-th/0002230) when taking a scalar field on an asymptotic AdS background in Fefferman-Graham coordinates $$ds^2 = \frac{l^2}{r^2}(dr^2 + G_{ij}(x, r) dx^i dx^j) $$ One can write a general asymptotic AdS spacetime in these coordinates as an expansion from the boundary $r \to \infty$, in which case on can write $$g(x, r) = g_{(0)} + r^2 g_{(2)} + \ldots + r^d g_{(d)} + h_{(d)} r^d \log r^2 + \mathcal{O}(r^{d+1}) $$ A scalar field on this background then looks, near the boundary at $r \to \infty$, as $$ \Phi(x, r) = r^{d- \Delta} \left( \phi_{(0)} + r^2 \phi_{(2)} + \ldots + r^{2\Delta - d} \phi_{2\Delta-d} + r^{2\Delta-d} \log r^2 \psi_{(2\Delta-d)} \right)+ \mathcal{O}(r^{\Delta+1}) $$ Here I was initially somewhat surprised to see this logarithmic term. I vaguely understand that since we are expanding around $r \to \infty$ we need to take into account all different sorts of asymptotic behaviour (and thus also have to include asymptotic logarithmic behaviour).

I can however not find any rigorous explanation of this. If I define $z = 1/r$ and write a general expansion for $\Phi$ as an expansion around $z = 0$, how do I obtain again the logarithmic behaviour when I go back to $r = 1/z$?.

Answer 1

The main theme here is Singularity Analysis which describes the asymptotic behavior of a large class of functions with moderate growth and decay and the strongly connected asymptotic behavior of the coefficients of their generating functions. At first we recall the simpler

Rational and meromorphic functions:

When analysing rational and meromorphic functions locally near a singularity $\zeta$, the main factors are of the form \begin{align*} \left(1-\frac{z}{\zeta}\right)^{-r}\qquad\qquad r\in\mathbb{Z}_{\geq 1} \end{align*} Their coefficients involve asymptotically exponential-polynomials, i.e. finite linear combinations of elements of type \begin{align*} \zeta^{-n}n^{r-1}\qquad\qquad r\in\mathbb{Z}_{\geq 1} \end{align*}

Functions with moderate growth and decay:

In singularity analysis we consider a much larger class of functions having other singularities than mere poles. The functions under consideration have as central elements of an expansion at a singularity $\zeta$ \begin{align*} \left(1-\frac{z}{\zeta}\right)^{-\alpha}\left(\log\frac{1}{1-\frac{z}{\zeta}}\right)^\beta\qquad\qquad \alpha,\beta\in\mathbb{C}\tag{1} \end{align*} These elements contribute asymptotically terms of type \begin{align*} \zeta^{-n}n^{\alpha-1}\left(\log n\right)^\beta \end{align*}

We see depending on $\alpha$ and $\beta$ we have either powers or logarithmic contributions or both parts to consider.

Table of functions:

To better see the relationship a few functions together with their coefficients, exact and asymptotically.

\begin{array}{lll} \text{function}\qquad&\qquad \text{coeff (exact)}\qquad & \quad\text{coeff (asymp.)}\\ \hline \\ 1-\sqrt{1-z}&\qquad\frac{2}{n4^n}\binom{2n-2}{n-1}&\qquad\sim\quad\frac{1}{2\sqrt{\pi n^3}}\\ \\ \frac{1}{\sqrt{1-z}}&\qquad\frac{1}{4^n}\binom{2n}{n}&\qquad\sim\quad\frac{1}{\sqrt{\pi n}}\\ \\ \frac{1}{1-z}&\qquad1&\qquad\sim\quad1\\ \\ \frac{1}{1-z}\log\frac{1}{1-z}&\qquad H_n&\qquad\sim\quad\log n\\ \\ \frac{1}{(1-z)^2}&\qquad n+1&\qquad\sim\quad n\\ \end{array}

Functions addressed by OP:

If a function has multiple dominant singularities of type (1) the separate contributions of each of the singularities are to be added up, resulting in asymptotic expansions given by OP.

Note: The basic facts mentioned here are from the classic Analytic Combinatorics (chap. VI) by P. Flajolet and R. Sedgewick which contains a detailed elaboration of singularity analysis.