How do I partial fraction this

Question

I have this fraction that I want to express as partial fractions:

$$\frac{s}{(s^2+1)(s-1)}$$ How do I do it?

I came as far as the expression:

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

But how do I solve this for A and B?

Answer 1

You're getting there.

What you're missing is that you'll need a first-degree polynomial for one of them rather than a constant, because of the quadratic (second-degree) in the bottom of one:

$$\frac{s}{(s^2+1)(s-1)} = \frac{A}{(s-1)} + \frac{Bs+c}{(s^2+1)}$$

Answer 2

You equation has no solution: you have to have $B=0$ to nullify the quadratic term, and then oyu run into another problem.

This is because you did write the wrong equation: try with $$ \frac s{(s^2+1)(s-1)} = \frac{As+B}{s^2 + 1} + \frac {C}{s-1} $$ instead

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alternative computation: once you have the good partial fractions, you can compute $A,B,C$ with another method:

• as $\lim_{s\to\infty}s \frac s{(s^2+1)(s-1)} = 1$ you get $A+C= 0$
• as $\lim_{s\to 1} (s-1) \frac s{(s^2+1)(s-1)} = 1/2$ so $C= 1/2$

etc.

Answer 3

Write s as $s=\frac{1}{2}(s^2+1-(s-1)^2)$