1. partial fraction decomposition

Hi

can someone help me with the following partial fraction decomposition. It is important that the answer has no complex number.

$\displaystyle \frac{4}{s^{2}(s^{2}+1)} = \frac{A}{s^{2}} + \frac{B}{s^{2}+1}$
is
$\displaystyle A(s^{2}+1) + Bs^{2} = 4$

2. Remember that when a factor has a multiplicity greater than one, you should list that factor along with all its powers. In other words, since $\displaystyle s^2 = s \cdot s$, we should have factors of $\displaystyle s$ and $\displaystyle s^2$.

Also, since $\displaystyle s^2+1$ cannot be factored any further, but has degree 2, we need to use two constants in the numerator for that term.

The whole thing should look like this:

$\displaystyle \frac{4}{s^{2}(s^{2}+1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{Cs+D}{s^2+1}$

By the way, a good check to make sure you didn't forget any terms is to make sure the number of constants you have should equal the degree of the original denominator.

3. Allright thanks.

$\displaystyle 4 = As(s^{2}+1) + B(s^{2}+1) + (Cs+D)s^{2}$
4 = A + B
0 = A
0 = B + D
0 = A + C

so
A=0, B=4, D=-4 and C=0

which seems to be correct.