Approximaths

Padé approximants: Convergence III

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For a function f(z) analytic at z=0, with Taylor series

\displaystyle f(z) = \sum_{k=0}^{\infty} a_k z^k

valid within its radius of convergence R, the Padé approximant of order [m/n], denoted \frac{P_m(z)}{Q_n(z)} with \deg P_m \leq m and \deg Q_n \leq n, satisfies

\displaystyle f(z) - \frac{P_m(z)}{Q_n(z)} = O(z^{m+n+1})

near z=0. The rational structure of \frac{P_m(z)}{Q_n(z)} allows it to approximate f(z) beyond the disk |z| < R by modeling singularities (e.g., poles or branch points) through the zeros of Q_n(z). This enables analytic continuation into regions where the Taylor series diverges.

Formally, for a meromorphic function f(z) in a domain D, the diagonal Padé approximants [n/n] often converge to f(z) in D \setminus S, where S is the set of poles of f:

Let f(z) be meromorphic in a domain D \subseteq \mathbb{C}, with a set of poles S of finite total multiplicity. The diagonal Padé approximants [n/n], defined as rational functions \frac{P_n(z)}{Q_n(z)} satisfying

\displaystyle f(z) - \frac{P_n(z)}{Q_n(z)} = O(z^{2n+1})

near z=0, converge uniformly to f(z) on compact subsets of D \setminus S as n \to \infty.

The zeros of Q_n(z) approximate the poles in S, enabling analytic continuation of f(z) beyond the radius of convergence of its Taylor series.

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