◄ WORLD V · SONNY 5DART 309 · a helldive at the net

THE PADÉ APPROXIMANT reconstruct a whole function — poles and all — from a few terms of its series

A power series only knows a function near one point and diverges past the nearest singularity. A Padé approximant reconstructs far more: it matches the same few series coefficients with a ratio of polynomials P(x)/Q(x), and because a rational function can have poles, it can recover the function's blow-ups and reach well beyond the series' radius of convergence. Truncate 1 + x + x² + … and its [1/1] Padé is exactly 1/(1−x) — pole and all, recovered from three coefficients. It is how physicists resum divergent perturbation series and how special functions are evaluated. Reconstruction of a global object from a local germ, caused by the rational structure that a polynomial cannot supply.

THE TECHNIQUE match the series with P(x)/Q(x); rational form recovers poles

The truncated series 1 + x + x². The demo forms its [1/1] Padé approximant and evaluates it beyond where the series converges: live demo







  
HISTORY & CREDIT Henri Padé, 1892

  
“A power series is all you can know from its coefficients.” — a Padé approximant reads more out of the same few coefficients: a rational form recovers the poles and reaches past the series' convergence. More function from the same data. cited

  
1892 · Henri Padé — the systematic theory of rational approximants (precursors in Frobenius, Cauchy, Jacobi).
the fact · the [1/1] Padé of Σ xⁿ is 1/(1−x) — the pole at 1, recovered.
now · resummation of divergent series in physics, special-function evaluation, control theory.

  
Match the coefficients with a ratio of polynomials and the poles a series must hide become explicit — the function recovered past its own radius of convergence. Padé 1892






  
RECOMMEND FOR I-13 function recovered from a truncated series, computed

  
On the canonical compiler, the [1/1] Padé of 1+x+x²+… is 1/(1−x); evaluated at x=0.5 it reconstructs 2 — the true value of the resummed function:

  
$ i13 run rec_pade.i13   # [1/1] Pade of 1+x+x^2+... = 1/(1-x)
pade = 2       -- reconstructs 1/(1-0.5) = 2, the function behind the series

  
Recommend: the Padé approximant is reconstruction of a whole function from a local germ, and i13 evaluates the recovered form: the [1/1] Padé of 1+x+x²+… is 1/(1−x), giving 2 at x=0.5 — the true resummed value, with the pole at x=1 now explicit. The load-bearing structure is rationality: a polynomial can never have a pole, so it cannot recover a blow-up; a ratio can, which is what lets three coefficients reconstruct a function that its own series only knew locally. It is the batch's most ambitious recovery — a global object from a handful of numbers — and, like all the rest, caused by the structure the reconstruction is allowed to use.