Approximaths

Stieltjes function I

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The function f(x) is a Stieltjes function if it has the following form:

\displaystyle f(x) = \int_{0}^{\infty} \frac{\rho(t)}{1+xt} \, dt

where \rho(t) is a ‘weight function’ with \rho(t) \geq 0 for all 0 \leq t < \infty and the moments of \rho(t) exist. The moments are defined as:

\displaystyle a_n = \int_{0}^{\infty} \rho(t) t^n \, dt

Using the geometric series expansion of \frac{1}{1 + xt}, the Stieltjes function has the following asymptotic expansion:

\displaystyle f(x) = \int_{0}^{\infty} \frac{\rho(t)}{1+xt} \, dt \sim \int_{0}^{\infty} \sum_{n=0}^{\infty} (-1)^n x^n t^n \rho(t) \, dt
\displaystyle = \sum_{n=0}^{\infty} (-1)^n \left( \int_{0}^{\infty} \rho(t) t^n \, dt \right) x^n
\displaystyle = \sum_{n=0}^{\infty} (-1)^n a_n x^n

This series is called a ‘Stieltjes series’ and is often divergent since the moments a_n can grow sufficiently rapidly for the series to have zero radius of convergence. Let begin the exploration of Stieltjes functions with an example. Let set \rho(t) = e^{-t}. We have:

\displaystyle f(x) = \int_{0}^{\infty} \frac{e^{-t}}{1+xt} \, dt
\displaystyle \sim \sum_{n=0}^{\infty} (-1)^n \left( \int_{0}^{\infty} e^{-t} t^n \, dt \right) x^n
\displaystyle = \sum_{n=0}^{\infty} (-1)^n n! x^n

So \rho(t) = e^{-t} \implies a_n = n!.

Stieltjes functions and their associated series offer a classic illustration of how divergent asymptotic expansions can still encode precise analytic information, a theme that continues to play a central role in asymptotic analysis and resurgence theory.

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