PARTIAL FRACTION Help!

1. Apr 20, 2010

wavingerwin

1. The problem statement, all variables and given/known data
How to break down:
$$\frac{1}{(s^{2}+1)^{2}}$$
into partial fractions?

2. Relevant equations
-

3. The attempt at a solution
I have tried:
$$\frac{1}{(s^{2}+1)^{2}} = \frac{1}{(1+i)^{2}\times(1-i)^{2}} = \frac{A}{(s+i)} + \frac{B}{(s+i)^{2}} + \frac{C}{(s-i)}} + \frac{D}{(s-i)^{2}}$$

and

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

and

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

but none of them works..

2. Apr 20, 2010

tiny-tim

Hi v_bachtiar!

Why isn't 1/(s2 + 1)2) good enough as it is?

But if you do want to break it down further, your first try should have worked …

what did you get?

3. Apr 20, 2010

wavingerwin

It is not good enough because I need to perform an inverse Laplace transform on the fraction.
And at my level, I only use tables and some basic theorems (convolution, shift in s etc.) and 1/(s2 + 1)2) is not on the table :(

I have attached my working using the first try..

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4. Apr 20, 2010

tiny-tim

Hi v_bachtiar!

What makes you think A B C and D are real?

Hint: to simplify it, what are 1/(s - i) ± 1/(s + i) and 1/(s - i)2 ± 1/(s + i)2 ?

5. Apr 20, 2010

wavingerwin

They are 2s/(s2+1) or 0
and
(2s2-2)/(s2+1)2 or 4si/(s2+1)2

so..

(As-Ai+Cs+Ci) / (s2+1) + (B(s+i)2+D(s-i)2) / ((s2+1)2) = 1

and As-Ai+Cs+Ci = 2s , B(s+i)2+D(s-i)2 = 2s2-2

is this right?

6. Apr 20, 2010

wavingerwin

oh, i mean:

(As-Ai+Cs+Ci) + (B(s+i)2+D(s-i)2) = 1

7. Apr 21, 2010

tiny-tim

Hi v_bachtiar!

(just got up :zzz: …)
No, 2s/(s2+1) or 2i/(s2+1)

ok, rewrite them as

(2s3+s)/(s2+1)2 or i(2s2+2)/(s2+1)2

Now can you see how to easily combine them with the others to get 1/(s2+1)2 ?

8. Apr 21, 2010

u can do as
A/s^2+1 + BX/(S^2+1)2

9. Apr 21, 2010

wavingerwin

hi tiny-tim,

you mean combine (add) them with (2s2-2)/(s2+1)2?

so 1 = (2s3+s) + (2s2-2)

then, where do I imply the A, B, C, and D?

(thank you for your help so far)

10. Apr 22, 2010

tiny-tim

How about (2s2+2) and (2s2-2) ?