2
$\begingroup$

I've been recently acquainted with a well known technique called " partial fraction decomposition" which allows, for example to express $\frac {x} {x^2-1}$ as $\frac {1}{2(x+1)} + \frac {1} {2(x-1)}$.

If I am correct, this technique is ordinarily used in calculus to integrate rational functions ( quotients of polynomials).

My question is simply whether there is some basic application of this technique outside integration problems, and even outside Calculus.

Is it possible to exhibit a possible use of PFD in algebra proper, and particularly, it would be nice to see a way it can be used in the discussion of a rational function.

Here I am not interested essentially in sophisticated applications, but rather in basic ones ( at "Algebra $2$" level, say).

Any reference is also welcome.

Thanks in advance.

$\endgroup$
6
  • 4
    $\begingroup$ For other examples, telescoping sums/series (1), Taylor expansions (2). $\endgroup$
    – dxiv
    Commented Dec 6, 2021 at 19:35
  • 2
    $\begingroup$ This can be used to prove general theorems about difference equations/recurrence relations (using @dxiv's idea of Taylor expansions - or binomial expansions): a good method for seeing what is going on when there are multiple roots of the characteristic equation. $\endgroup$
    – Mark Bennet
    Commented Dec 6, 2021 at 21:03
  • $\begingroup$ Finding the Egyptian fraction expansion of a fraction may use some PFD. $\endgroup$
    – soupless
    Commented Jan 18, 2022 at 8:16
  • 1
    $\begingroup$ In signal processing for example, you often use partial fraction decomposition in the computation of inverse Laplace transforms. Even though, it's technically kind of related to integration, I think they are quite relevant to mention here :p $\endgroup$
    – seboll13
    Commented May 14, 2022 at 8:01
  • 1
    $\begingroup$ Here is an application of partial fraction expansion, in the field of euclidean geometry. If $P\in\mathbb{C}[X]$ has degree $n\geqslant 2$, then the roots of $P'$ (the dérivative of $P$) belong to the convex hull of the set of roots of $P$. This result is known as Luca's theorem. See here : en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem $\endgroup$
    – Adren
    Commented May 14, 2022 at 8:33

0

Reset to default

You must log in to answer this question.