Padé approximant

Summary

A Padé approximant of a power series is a rational function with prescribed degrees. Its own power series agrees with the given one to as high an order as these degrees allow. For numerator degree at most $m$ and denominator degree at most $n$, the agreement is up to the power $z^{m+n}$. Padé approximants often approximate a function far beyond the disc of convergence of its Taylor series. They can also show the location of poles. They are closely linked to continued fractions and to orthogonal polynomials.