Notes on Whittaker & Watson, Chapter VIII

Whittaker & Watson, A Course of Modern Analysis. Chapter VIII, Asymptotic Expansions and Summable Series.

8.1. An example: the function $f(x)=\int_x^\infty t^{-1}e^{x-t}$ can be expanded as $$ f(x)=\sum_{m=0}^n\frac{(-1)^mm!}{x^{m+1}} + (-1)^{n+1}(n+1)!\int_x^\infty \frac{e^{x-t}}{t^{n+2}}dt. $$ The series sum $S_n=\sum_{m=0}^n\frac{(-1)^mm!}{x^{m+1}}$ diverges, and the remainder $|f(x)-S_n(x)|<\frac{1}{2^{n+1}n^2}$ for $x\ge 2n$.

8.2 The Poincaré’s definition of asymptotic expansion: $$ f(x)\sim A_0+\frac{A_1}{z}+\frac{A_2}{z^2}+\cdots+\frac{A_n}{z^n}+\cdots $$ The divergent series on the RHS is the asymptotic expansion of $f(x)$ if the remainder $R_n(z)=z^n\left(f(z)-S_n(z)\right) \to 0$ when $|z|\to\infty$ given $n$, even if $\lim_{n\to\infty}R_n(z) = \infty$ at fixed $z$.

8.3 Multiplication of asymptotic expansions. If $$ \begin{gather*} f(z)\sim \sum_{m=0}^\infty A_mz^{-m},\newline \phi(z)\sim \sum_{m=0}^\infty B_mz^{-m}, \end{gather*} $$ then $f(z)\phi(z)\sim \sum_{m=0}^\infty C_mz^{-m}$ with $C_m=A_0B_m+A_1B_{m-1}+\cdots+A_mB_0$.

8.31 Integration. If $f(x)\sim\sum_{m=0}^\infty A_mz^{-m}$, $$ \int_x^\infty f(x)dx\sim\sum_{m=2}^\infty\frac{A_m}{(m-1)x^{m-1}}. $$

8.32 Uniqueness of an asymptotic expansion.

8.4 ‘Summing’ series.

8.41 The ‘Borel sum’ of $\sum_{n=0}^\infty a_nz^n$ is the analytic continuation by Borel’s integral $\int_0^{\infty}e^{-t}\phi(tz)dt$ with $\phi(tz)=\sum_{n=0}^\infty \frac{a_nt^nz^n}{n!}$.

8.42 Euler’s sum (Abel’s) for $\sum_{n=0}^\infty a_n$ is defined as $$ \lim_{x\to1-0}\sum_{n=0}^\infty a_nx^n. $$ 8.43 Cesàro summation $(C, 1)$ for $\sum_{n=0}^\infty a_n$ $$ \lim_{n\to\infty}\frac{1}{n}(s_1+s_2+\cdots+s_n), $$ where $s_n$ is the partial sum.

8.44 Riesz $$ \lim_{\nu\to\infty}\sum_{n=1}^\nu\left(1-\frac{\lambda_n}{\lambda_\nu}\right)^ra_n $$ with $\lambda_n\to\infty$.

8.5 (Hardy) Tauberian theorem for Cesàro summability: If $\sum_{n=0}^\infty a_n$ is summable (Cesàro) and $a_n=O(1/n)$, then the series $\sum_{n=0}^\infty a_n$ converges.

[A proof is outlined in Rudin, Principles of of Mathematical Analysis, Ch. 3, Exercise 14.]