What is Taylor Expansion for ( S)/((S^2+??^2 )(S^(2-?)+??^2 S^?))

Question

i have tried adding and subtract  ω_º^2 for the second term 

the final answer should be 

((∑_k,q(-1)^k+q ω_º ^2 k ω^2(k+q) (k+q)!) /(k! q!)) *(S^β /( S ²+ω_º ²)^K+Q+1),Where β= -1-2 k+α+2α k+2 α q

Answer 1

I figured out the general idea for this problem:

1. Find the inverse Laplace transform f(t) of the fraction F(s) you are given. This will involve using partial fraction decomposition and the shifting property. Depending on the value of α, you will get sines and cosines, multiplied by powers of t.

2. Find the Taylor series of f(t), which should be elementary.

3. Find the Laplace transform F(s) of this Taylor series, which will give you the answer you have.

 

Here's a simple example that doesn't require partial fractions and hence no double sum. Let's say

 

F(s) = 1/(s²+ω²).

 

Then

 

f(t) = sin(ωt) = ∑_k (-1)^k (ωt)^2k+1/(2k+1)!

F(s)= ∑_k (-1)^k ω^2k+1 /s^2k+2.

 

For a more detailed answer, please check all your terms.