Padé approximant

Remarks

Existence: condition 3 for the coefficients of $z^{m+1}, \dots, z^{m+n}$ is a system of $n$ homogeneous linear equations in the $n + 1$ coefficients of $Q$. So a nonzero $Q$ exists, and then $P$ is the truncation of $Q f$ at degree $m$. Uniqueness: for two pairs, $P Q' - P' Q = Q'(Q f - P) - Q(Q' f - P')$ is a polynomial of degree at most $m + n$ that is $O(z^{m+n+1})$, so it is $0$. Example: for $f = e^z$, $m = n = 1$, take $Q = 1 - z/2$ and $P = 1 + z/2$. Then $Q f = 1 + z/2 + 0 \cdot z^2 - z^3/12 + \cdots$, so $Q f - P = O(z^3)$ and $[1/1]_{e^z} = (2 + z)/(2 - z)$. Non-example: $P = 1 + z$ and $Q = 1$ satisfy clauses 1 and 2 for $m = n = 1$, but $Q f - P = z^2/2 + \cdots$ fails clause 3. For $n = 0$ the approximant $[m/0]_f$ is the Taylor polynomial of degree $m$. When $Q(0) \ne 0$ one usually normalizes $Q(0) = 1$, and then $f - P/Q = O(z^{m+n+1})$; when $Q(0) = 0$ the approximant may match fewer coefficients.