Inverse laplace and power series

[oddiseas]

Homework Statement

I am trying to figure out how to represent an inverse laplace transform by a power series. There is an example in my book but it is not very well explained.

f(s)=1/s+1 which i know is the transform of y=e^-t.

In the book they use the fact that L(t^n)= n!/s^n+1. and therefore a taylor series representation given by t^n/n!=the inverse of 1/s^n+1. Therefore our power series in s has this form. After this point i am totally lost. They state that 1/s+1 =1/s(1+1/s).
and the solution is therefore (1/s)-(1/s^2)+(1/s^3) etc.

Now id like to know, WHY do they take the factor of s out of the equation?
ANd then how do they find the coefficients of the s terms? do they differentite f(s) to find the coefficient of each s term? like we do for a taolr series
Is there an easier way?
and waht value of s, is it evaluated at to find the coefficients?
If someone understands this i would appreciate an explanation, because this book seems to always assume that the reader is a 20 year mathematics veteran or something!
So please don't just post the answer because i already know the answer, i am trying to understand this concept.

Homework Equations

The Attempt at a Solution

 

[mighty2000]

it is important to understand the concepts and theories behind mathematical equations rather than just blindly using them. In this case, the inverse Laplace transform by power series is a method used to find the inverse Laplace transform of a function by expressing it as a series of powers of s. Let's break down the steps involved in this process to gain a better understanding.

First, we start with the given function f(s)=1/s+1. We know that this is the Laplace transform of y=e^-t. The inverse Laplace transform is defined as the integral of the function multiplied by e^st, where s is the variable in the Laplace domain. Therefore, we can write the inverse Laplace transform of f(s) as:

L^-1{f(s)} = ∫(f(s)e^st)ds

Now, to represent the inverse Laplace transform by power series, we need to express f(s) as a series of powers of s. This is where the Taylor series representation comes into play. In the book, they have used the fact that L(t^n)= n!/s^n+1. This means that for a function of the form t^n, its Laplace transform is n!/s^n+1.

Using this, we can express the given function f(s) as a series of powers of s:

f(s) = 1/s+1 = 1/s(1+1/s) = 1/s(1+1/s^2+1/s^3+...+1/s^n+...)

We have taken out the factor of s from the equation because it helps us in finding the coefficients of the s terms. Now, to find the coefficients, we need to differentiate f(s) with respect to s. This is similar to finding the coefficients in a Taylor series. For example, to find the coefficient of s^2, we differentiate f(s) twice and evaluate it at s=0. This will give us the coefficient of s^2 as 2!. Similarly, for s^n, we differentiate f(s) n times and evaluate it at s=0 to get the coefficient of s^n as n!.

So, the coefficients in the power series representation of f(s) are determined by differentiating f(s) and evaluating it at s=0.

In summary, the inverse Laplace transform by power series is a method of representing the inverse Laplace transform of