# Inverse Laplace transform for small 's', Taylor expansion

• perr
In summary, the conversation discusses a question about the Laplace transform of a Taylor series expansion and the inverse Laplace transform of a given function. The person is seeking help on how to solve this problem and suggests writing and expanding the function before taking the inverse Laplace transform. They also provide a link for more information.
perr
Dear all,

This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem:

Consider the Laplace transform

F(s) = 1 / ( s - K(s) ) ,

where

K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s ) ].

See PDF-attachment. ( 'Lambda' and 'b' are real numbers, typically 'Lambda=1000' and 'b=10'). I want to explore the inverse Laplace transform of this F(s) for large time (t-> infinity), i.e. s->0.

How can this be done?

Perhaps write/expand F(s) on the form F(s) = A/s + B + C*s + D*s^2 + ... , and then take inverse Laplace transform of each these terms?

I appreciate any help!

perr

#### Attachments

• 21 Inverse Laplace, (9. juli 2011).pdf
80.8 KB · Views: 357

## 1. What is the definition of an inverse Laplace transform for small 's'?

The inverse Laplace transform for small 's' refers to the process of finding the original function from its Laplace transform, where the value of 's' is close to zero. This can be represented mathematically as s → 0.

## 2. How is the Taylor expansion used in inverse Laplace transforms for small 's'?

The Taylor expansion is a mathematical technique used to approximate a function by evaluating its derivatives at a single point. In the context of inverse Laplace transforms for small 's', the Taylor expansion is used to approximate the original function by expanding it around the point where s = 0.

## 3. What is the significance of small 's' in inverse Laplace transforms?

Inverse Laplace transforms for small 's' are important because they allow us to analyze the behavior of a function as 's' approaches zero, which can provide insights into the long-term behavior of the original function. It is also useful when dealing with systems that have a stable equilibrium point at s = 0.

## 4. Are there any limitations to using Taylor expansion for inverse Laplace transforms for small 's'?

Yes, there are limitations to using Taylor expansion for inverse Laplace transforms for small 's'. One limitation is that it only provides an approximation of the original function, which may not be accurate for all values of 's'. Additionally, the accuracy of the approximation depends on the number of terms included in the Taylor expansion.

## 5. Can inverse Laplace transforms for small 's' be used for all types of functions?

No, inverse Laplace transforms for small 's' are generally used for functions that have a finite limit as 's' approaches zero. If the function does not have a finite limit, then other methods, such as partial fraction expansion, may be more appropriate for finding the inverse Laplace transform.

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