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?

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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&#039;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?

Cicolinif73 · Accepted Answer

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)=∑Nn=0anxn1+∑Mm=1bmxm, 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: a0,a1,…aN, along with b1,…,bM. So, we assert that the rational function should match the function we are approximating all the way up to M+Nth order around some point x0: f(x0)=R(x0) dfdx(x0)=dRdx(x0) d2fdx2(x0)=d2Rdx2(x0) dM+NdxM+N(x0)=dM+NRdxM+N(x0) It seems natural that this would uniquely specify what R must be, and indeed it does.

If I have an analytic function of a complex variable, I can write a Taylor series and in some cases

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