The partial-fraction expansion for F(s) = N(s)/D(s) can be found using the MATLAB function [K, p, k]= residue (N, D) where K= residue, p=roots of denominator, and k= direct quotient. Write a MATLAB script using the residue function to find the partial-fraction expansion of F(s), and determine the inverse Laplace transform f(t) (DO NOT USE MATLAB TO FIND f(t). Use Laplace Transform Table TO CONVERT F(s) to f(t)). a. F(s)= b. F(s)= 2(s+3)(s+5)(s+7) s(s+8)(s² +10s +100) (7s² +9s +12) s(s+7)(s² +10s +100)

Given:

a. $F\left(s\right) = \frac{2(s+3)(s+5)(s+7)}{s(s+8)(s^2+10s+100)}$

b. $F\left(s\right) = \frac{(7s^2+9s+12)}{s(s+7)(s^2+10s+100)}$

Inverse Laplace Transform:

The inverse Laplace transform is an operation that is used to convert a function from the Laplace domain to the time domain. If F(s) is the Laplace transform of a function f(t), denoted by F(s) = L[f(t)], then the inverse Laplace transform of F(s) is given by:

$f\left(t\right)=\frac{1}{2\pi i}\int \gamma e^{st}F\left(s\right)ds$

where γ is a contour in the complex s-plane that lies to the right of all singularities of F(s). The integral is carried in the counterclockwise direction.

In practice, finding the inverse Laplace transform involves the use of tables, partial fraction decomposition, and contour integration techniques. The choice of the contour γ depends on the nature of the singularities of F(s) and the behaviour of f(t) as t → ∞.

The inverse Laplace transform is used in many areas of science and engineering, such as control theory, signal processing, and differential equations. It allows us to study the time-domain behaviour of systems that are described by Laplace-transformed equations and to design controllers and filters that operate in the frequency domain.