Inverse Laplace transform for small 's', Taylor expansion

AI Thread Summary
The discussion focuses on finding the inverse Laplace transform of the function F(s) = 1 / (s - K(s)), where K(s) involves a logarithmic term dependent on real parameters Lambda and b. The user seeks to analyze F(s) as s approaches zero, particularly for large time limits (t → ∞). A proposed method includes expanding F(s) into a series form, such as F(s) = A/s + B + C*s + D*s^2, and then applying the inverse Laplace transform to each term. The user references relevant Wikipedia articles for additional context on Laplace transforms. The goal is to derive insights into the behavior of the system as time progresses.
perr
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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
 

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