aanpalendmw

2022-07-22

partial fraction decomposition of $\frac{{k}^{4}}{\left(a\phantom{\rule{thinmathspace}{0ex}}{k}^{3}-1{\right)}^{2}}$
I have to perform complex partial fraction decomposition of the following term:
$\frac{{k}^{4}}{\left(a\phantom{\rule{thinmathspace}{0ex}}{k}^{3}-1{\right)}^{2}}$
where $a$ is a real positive number.
and I would like to know if it is possible to reduce it to a sum of fractions of the type $\frac{A}{k±z}$,$\frac{B\phantom{\rule{thinmathspace}{0ex}}k}{{k}^{2}±z}$,$\frac{C\phantom{\rule{thinmathspace}{0ex}}k}{{k}^{2}-y}$ or similar. Where $z$ is a complex number and $y$ is a real number.
If it is not possible to reduce it to the kind of fraction I listed above other type of decomposition might work too.
Any hint on the process, or any reference would be nice.

yelashwag8

I would start by making the change of variable , which transforms the expression into

The denominator is now factored. Given that there is only a two-cube difference, this is simple.

The quadratic is irreducible over the reals, as you can see. Then (ignoring the constant ) we have

Glenn Hopkins

Set and denominator will be factored as

After that, you split into partial fractions.

where  are unknowns and must be obtained solving a six equations linear system. Prior to considering the numerator of the result, you first sum all the fractions.

Numerator must be identical to so coefficients (from degree 0 up to 5 must be
The system offers several lovely solutions.

Hence, the initial fraction can be expressed as

with

Do you have a similar question?