If I have an analytic function of a complex variable, I can write a Taylor series and in some cases can truncate the high powers to obtain a good approximation over some part of the function's domain. I would like to be able to generate rational functions (quotients of polynomials, as in $n(x)/d(x)$ where $n$ and $d$ are polynomial functions) that play a similar role. Is there a general way to do this?
This question is motivated by an informal observation that the functions I am trying to approximate "look like" rational functions.
Best Answer
This describes a Padé approximant. To see how it arises, first note that in a rational function of degree $N$ in the numerator and degree $M$ in the denominator,
$$ R(x) = \frac{\sum^N_{n=0}a_n x^n}{1 + \sum^M_{m=1}b_m x^m}, $$ where a $1$ has been taken out of the sum in the denominator to eliminate a redundancy in the coefficients, there are $M+N+1$ free parameters: $a_0,a_1, \cdots a_N$, along with $b_1,\cdots,b_M$. So, we assert that the rational function should match the function we are approximating all the way up to $M+N$th order around some point $x_0$: $$ f(x_0) = R(x_0) $$ $$ \frac{df}{dx}(x_0) = \frac{dR}{dx}(x_0) $$ $$ \frac{d^2f}{dx^2}(x_0) = \frac{d^2R}{dx^2}(x_0) $$ $$ \vdots $$ $$ \frac{d^{M+N}f}{dx^{M+N}}(x_0) = \frac{d^{M+N}R}{dx^{M+N}}(x_0) $$
It seems natural that this would uniquely specify what $R$ must be, and indeed it does.