Inverse Laplace Transform

  • Thread starter NT123
  • Start date
  • #1
NT123
28
0

Homework Statement

Calculate the inverse Laplace transform of exp(-as)/s (a is a constant).



Homework Equations





The Attempt at a Solution

I need to calculate the integral (1/2*pi*i)int(c-i(inf), c+i(inf))(exp(s(t-a))/s).

I'm guessing I need to integrate around a circular contour centred at c, with a sufficiently large radius to contain zero.

The pole is at zero so I guess the residue is just exp(s(t-a))(0) = 1. I'm not sure how to show the rest of the integral ---> 0. Any help would be appreciated.
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,568
774
What about using the time shifting formulas?
 
  • #3
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,758
2,394

Homework Statement



Calculate the inverse Laplace transform of exp(-as)/s (a is a constant).

The Attempt at a Solution



I need to calculate the integral (1/2*pi*i)int(c-i(inf), c+i(inf))(exp(s(t-a))/s).

I'm guessing I need to integrate around a circular contour centred at c, with a sufficiently large radius to contain zero.
Not quite. The integral is along the line Re(s)=c where c must be positive so that the line will be in the region of convergence. To use the residue theorem, you have to close the contour, and the choice of how to close it will depend on whether t>a or t<a.
I'm not sure how to show the rest of the integral ---> 0. Any help would be appreciated.
The complete contour consists of two parts: the line from the original integral and the piece you need to form a closed contour. Show that the integrand vanishes on that second piece.
 

Suggested for: Inverse Laplace Transform

  • Last Post
Replies
3
Views
414
  • Last Post
Replies
10
Views
738
Replies
7
Views
229
Replies
1
Views
271
Replies
5
Views
446
  • Last Post
Replies
2
Views
681
Replies
9
Views
545
  • Last Post
Replies
4
Views
597
Replies
3
Views
508
Replies
2
Views
362
Top