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Is it possible to do partial fraction expansion with specific degree of the variable?

Mathematica Asked by FillLost on July 23, 2021

For example, here is a rational polynomial:
$$frac{8 z^2}{8 z^2-6 z+1}$$

If I using Apart directly, the result is:
$$-frac{1}{4 z-1}+frac{2}{2 z-1}+1$$
but if I want to using $z^{-1}$ as the variable, the partial fraction expansion should be:
$$frac{-1}{1-frac{1}{4}z^{-1}}+frac{2}{1-frac{1}{2}z^{-1}}$$
Is this possible in Mathematica?

One Answer

What about

Apart[(8 z^2)/(8 z^2 - 6 z + 1) /. z -> 1/zi, zi] /. zi -> 1/z
(*4/(-4 + 1/z) - 4/(-2 + 1/z)*)

Correct answer by Ulrich Neumann on July 23, 2021

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