# Having some issues with partial fraction decomposition

Im having some trouble when I apply partial fraction decomposition on $$\frac{s^{2}+1}{s^{3}-s}$$ It can be simplified to $\frac{s^{2}+1}{s(s-1)(s+1)}$. Isn't the aim after this step to rewrite the expression into $$\frac{s^{2}+1}{s(s-1)(s+1)}=\frac{A}{s}+\frac{B}{s-1}+\frac{C}{s+1}$$?

It doesnt seem to work when I find the values for $A,B,C$. Is there something I have missed that maybe should be included in the numerators or?

• can you show what you have tried? – mathreadler Dec 19 '17 at 16:46
• As you haven't given us your calculation, I'm going to assume you have made an arithmetic error somewhere. – Lord Shark the Unknown Dec 19 '17 at 16:47
• $A + B + C = 1, B-C = 0, -A = 1$ – Doug M Dec 19 '17 at 16:49
• Yeah you guys are right, missed a minus sign in one of the coefficients. But im a bit confused to, is there some case in partial fraction decomposition that you have to have the variable in the numerator also? Or is it always constant coefficients in the numerators? For example $$\frac{s-2}{(s-1)(s^{2}+1)}$$ can this expression also be written on the form $\frac{A}{s-1}+\frac{B}{s^{2}+1}$ ? – Biggiez Dec 19 '17 at 16:53
• Based on the "$s$" I conjecture you're taking a course in differential equations and you're doing the Laplace transform right now. If so you really need a calculus book. – David C. Ullrich Dec 19 '17 at 17:04

$$\frac{s^{2}+1}{s(s-1)(s+1)}=\frac{A}{s}+\frac{B}{s-1}+\frac{C}{s+1}$$

$$\implies s^2 +1 = A(s^2 -1) + B(s^2+s) + C(s^2 - s)$$

Letting $s = 1$ gives $B=1$, letting $s = -1$ then gives $C =1$, and we then substitute to get $A = -1$.

You get by identifying coefficients for each monomial:$$\cases{\phantom{-}A+B+C=1\\\phantom{-A}+B-C=0\\-A\phantom{+B-C}\,\,\,=1}$$

This is a linear equation system we can write on matrix form:

$$\left[\begin{array}{rrr|r}1&1&1&1\\0&1&-1&0\\-1&0&0&1\end{array}\right]$$

If we solve it we see we get $[-1,1,1]^T$ corresponding to $A=-1,B=C=1$