Proving the Vanishing Integral in Inverse Laplace Transform by Residue Method

Click For Summary
To find the inverse Laplace transform of s/(s^2+a^2) using the residue method, it is necessary to prove that the integral over the contour not related to the desired integral vanishes as R approaches infinity. The discussion highlights the calculation of residues but emphasizes the challenge of demonstrating that the non-linear segment of the contour tends to zero. It is suggested that methods similar to those used in the proof of Jordan's theorem should be applied. The behavior of the integrand at large s is noted to be like 1/s, complicating straightforward estimation. Understanding these aspects is crucial for successfully completing the inverse transform.
NT123
Messages
26
Reaction score
0

Homework Statement

I need to find the inverse Laplace transform of s/(s^2+a^2), where a is a constant, by the method of residues. I need to prove the part of the contour not actually relating to the desired integral tends to zero as R---> infinity.



Homework Equations





The Attempt at a Solution

I have calculated the residues, however I am having trouble proving the integral of the contour which isn't the straight line from -R to R vanishes. Any help would be appreciated.
 
Physics news on Phys.org
You need to use the same methods here that are used in the proof of Jordan's theorem. The large s behavior is like 1/s, while the radius of the circle will be proportional to s, so a more straightforward way of estimating the integrand won't do.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

What is the Inverse Laplace Transform of e^(-sx^2/2)?
  • · Replies 10 ·
Replies
10
Views
2K
A tricky inverse Laplace transform
  • · Replies 8 ·
Replies
8
Views
2K
Does the Laplace Transform of e^(at)/t Exist?
  • · Replies 4 ·
Replies
4
Views
3K
Are these two inverse Laplace transform solutions equivalent?
  • · Replies 2 ·
Replies
2
Views
2K
How Do You Select Sigma for Different Regions in Inverse Laplace Transforms?
  • · Replies 5 ·
Replies
5
Views
2K
Inverse Laplace transform for an irreducible quadratic?
  • · Replies 2 ·
Replies
2
Views
3K
How to solve this problem using laplace transform?
  • · Replies 4 ·
Replies
4
Views
3K
Laplace transform applied to a differential equation
  • · Replies 1 ·
Replies
1
Views
1K
How Do You Solve This Inverse Laplace Transform Equation?
  • · Replies 2 ·
Replies
2
Views
2K
Inverse laplace transform (polynomial division? Complex roots?)
  • · Replies 5 ·
Replies
5
Views
3K