Find the constants in partial fractions?

Questions
1

I have this quotiënt split in partial fractions as following :

(s^2 + 3) / ((s - 1)^2 (s + 1)) ==  A1 / (s - 1) + A2 / (s - 1)^2  + A3 / (s + 1)

Is there an easy way to find the constants ? I tried several things with Solve but without succes.

Thank you very much

2

Have you looked at Apart?

Apart[(s^2 + 3)/((s - 1)^2 (s + 1))] (* 2/(-1+s)^2+(1+s)^(-1) *)
3

eq1 = (s^2 + 3)/((s - 1)^2 (s + 1)) == A1/(s - 1) + A2/(s - 1)^2 + A3/(s + 1)

sol = SolveAlways[eq1, s][[1]]

(* {A2\[Rule]2,A3\[Rule]1,A1\[Rule]0}*)

Simplify[eq1 /. sol]
4

Thank you both very much. I have found a solution myself but with more ‘hassle’ around it then yours.

f[s_] := (s^2 + 3) / ((s - 1)^2 (s + 1)) == A1 / (s - 1) + A2 / (s - 1)^2 + A3 /( s + 1)

Solve[ { f[s] /. s → 0, f[s] /. s → 2, f[s] /. s → 3}, {A1, A2, A3}]

{{A1 → 0, A2 → 2, A3 → 1}}

Greetings,Chiel Geeraert (Netherlands)