Month: October 2026

Padé to the Rescue

Recall that in the previous posts we attempted to calculate the following Stieltjes integral using perturbation techniques:

$\displaystyle I(\epsilon) = \int_{0}^{\infty} \frac{e^{-t}}{1 + \epsilon^2 t}\,dt$

For $\epsilon = 1$ , we obtain the target integral $I(1) = \int_{0}^{\infty} \frac{e^{-t}}{1+t}\,dt$ . Expanding $(1 + \epsilon^2 t)^{-1}$ as a formal power series and integrating term by term leads to the series:

$\displaystyle \sum_{k=0}^{\infty} (-1)^{k} k! \, \epsilon^{2k}$

This series is divergent for $\epsilon = 1$ and does not allow us to directly approximate the exact value of the integral. We can, however, approximate the exact value using Padé approximants. By making the change of variable $x = \epsilon^2$ , the series transforms into a dense power series in $x$ , $\sum_{k=0}^{\infty} (-1)^{k} k! \, x^k$ . We then construct its diagonal Padé approximants $[N/N](x)$ evaluated at $x = 1$ .

The following R code calculates and evaluates the $[5/5]$ Padé approximant using the pracma package (note: coefficients must be supplied in decreasing order of powers for pracma::pade):

library(pracma)

# Series coefficients a_k = (-1)^k * k! in DECREASING order of powers
k <- 10:0
p1 <- (-1)^k * factorial(k)

# Compute [5/5] Pade approximant with respect to x = epsilon^2
Q  <- pade(p1, d1 = 5, d2 = 5)
r1 <- Q$r1
r2 <- Q$r2

# Rational function evaluation
f1 <- function(x) polyval(r1, x) / polyval(r2, x)

# Evaluate at epsilon = 1 (x = 1^2 = 1)
f1(1)

# Plot graph
xs <- seq(-1, 1, length.out=100)
ys1 <- f1(xs)
plot(xs, ys1, type = "l", col="blue", xlab="x", ylab="P[5,5](x)")
grid()

In the table below, the values of successive diagonal Padé approximants $[N/N](x)$ evaluated at $x = 1$ are displayed. We can observe that these values converge rapidly towards the exact value of the integral ( $I \approx 0.596347$ ).

Padé approximants have therefore made it possible to evaluate a Stieltjes integral represented by a divergent series. This result can be extended to the class of all Stieltjes functions, which will be the subject of the next posts

Padé [N/N]	Padé approximant	Approximation at x = 1	Error
[1/1]	(1 + x)/(1 + 2x)	0.666667	7.03 × 10^-2
[2/2]	(1 + 5x + 2x²)/(1 + 6x + 6x²)	0.615385	1.90 × 10^-2
[3/3]	(1 + 11x + 26x² + 6x³)/(1 + 12x + 36x² + 24x³)	0.602740	6.39 × 10^-3
[4/4]	(1 + 19x + 102x² + 154x³ + 24x⁴)/(1 + 20x + 120x² + 240x³ + 120x⁴)	0.598802	2.46 × 10^-3
[5/5]	(1 + 29x + 272x² + 954x³ + 1044x⁴ + 120x⁵)/(1 + 30x + 300x² + 1200x³ + 1800x⁴ + 720x⁵)	0.597383	1.04 × 10^-3
[10/10]	(exact rational of degree 10)	0.596379	3.15 × 10^-5

October 1, 2026