Notes on Whittaker & Watson, Chapter VII, part 3

Whittaker & Watson, A Course of Modern Analysis. Chapter VII, The Expansion of Function in Infinite Series.

7.8 Borel’s theorem. For the Taylor series of $f(z)=\sum_{n=0}^\infty a_nz^n$ converging when $|z|\le r$, we define

$$ f_1(z)=\int_0^\infty e^{-t}\phi(zt)dt, $$ with $\phi(z)=\sum_{n=0}^\infty \frac{a_nz^n}{n!}$ being Borel’s function associated with $\sum_{n=0}^\infty a_nz^n$. The function $f_1$ thus defined agrees with $f$ when $|z|< r$.

7.81 Borel’s integral as analytic continuation to Borel’s polygon. For any $\zeta$ in the interior of the polygon, $\phi(\zeta t)=\frac{1}{2\pi i}\sum_{n=0}^\infty\frac{\zeta^nt^n}{n!}\int\frac{f(z)}{z^{n+1}}dz$ integrating on a circle with the segment from $0$ to $\zeta$ lying in the interior of its diameter. The sum of the integrand converges uniformly, thus

$$ \phi(\zeta t)=\frac{1}{2\pi i}\int z^{-1}f(z)\exp(\zeta tz^{-1})dz $$ which is bounded by $|\phi(\zeta t)|<F(\zeta) e^{\lambda t}$ where $\lambda$ is the maximum attained by the real part of $\zeta/z$ on the circle. The $z\to\frac{z}{\zeta}$ maps the contour to a circle with the diameter containing the real interval $[0,1]$. Therefore the maximum $\lambda<1$, and consequently the integral $\int_0^\infty e^{-t}\phi(\zeta t)dt$ is analytic at $t$.

7.82 Expansions in a series of inverse factorials. Using the results from 7.8 on the series $f(z)=\sum_{n=0}^\infty a_nz^{-n}$ with $|z|>r$, we have $f(z)=\int_0^\infty ze^{-tz}\phi(t)dt$ where $\phi(t)=\sum_{n=0}^\infty a_nt^n/n!$. With the change of variable $\xi=1-e^{-t}$ and $F(\xi)=\phi(t)$, we have $$ \begin{align*} f(z)&=\int_0^1z(1-\xi)^{z-1}F(\xi)d\xi\newline &= b_0+\frac{b_1}{z+1}+\frac{b_2}{z+2} + \cdots + \frac{b_n}{(z+1)(z+2)\cdots(z+n)}+R_n, \end{align*} $$ by repeated integrations by parts, where $b_n=F^{(n)}(0)$.