Compute the partial fraction decomposition of the following:
1/(s^2-1)
Factor the denominator into linear and irreducible quadratic terms:
1/((s-1) (s+1))
Then the partial fraction expansion is of the form:
1/((s-1) (s+1)) = θ_1/(s-1)+θ_2/(s+1)
Multiply both sides by (s-1) (s+1) and simplify:
1 = θ_1 (s+1)+θ_2 (s-1)
Expand and collect in terms of powers of s:
1 = θ_1-θ_2+(θ_1+θ_2) s
Equate coefficients on both sides, yielding 2 equations in 2 unknowns:
1 = θ_1-θ_2
0 = θ_1+θ_2
In matrix form the system is written as:
(1 | -1
1 | 1)(θ_1
θ_2) = (1
0)
In augmented matrix form, the system is written as:
(1 | -1 | 1
1 | 1 | 0)
Subtract row 1 from row 2:
(1 | -1 | 1
0 | 2 | -1)
Divide row 2 by 2:
(1 | -1 | 1
0 | 1 | -1/2)
Perform back substitution on the augmented upper-triangular matrix:
(1 | -1 | 1
0 | 1 | -1/2)
Add row 2 to row 1:
(1 | 0 | 1/2
0 | 1 | -1/2)
Read off the solutions:
θ_1 = 1/2
θ_2 = -1/2
Therefore:
Answer: |1/(s^2-1) = 1/(2 (s-1))-1/(2 (s+1))
