Skip to content Skip to navigation

Connexions

You are here: Home » Content » The Inverse Laplace Transform: Complex Integration

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

The Inverse Laplace Transform: Complex Integration

Module by: Steven Cox

Summary: (Blank Abstract)

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

The Inverse Laplace Transform

If qq is a rational function with poles { λ j |j=1…h} λ j j 1 … h then the inverse Laplace transform of qq is

ℒ-1qt≡12πⅈ∫Cqzⅇztdz ℒ q t 1 2 z C q z z t (1)
where CC is a curve that encloses each of the poles of qq. As a result
ℒ-1qt=∑j=1hres λ j ℒ q t j 1 h res λ j (2)
Let us put this lovely formula to the test. We take our examples from discussion of the Laplace Transform and the inverse Laplace Transform. Let us first compute the inverse Laplace Transform of qz=1z+12 q z 1 z 1 2 According to Equation 2 it is simply the residue of qzⅇzt q z z t at z=-1 z -1 , i.e., res-1=limz→-1ddzⅇzt=tⅇ-t res -1 z -1 z z t t t This closes the circle on the example begun in the discussion of the Laplace Transform and continued in exercise one for chapter 6.

For our next example we recall ℒ x 1 s=0.19s2+1.5s+0.27s+1/64s3+1.655s2+0.4978s+0.0039 ℒ x 1 s 0.19 s 2 1.5 s 0.27 s 16 4 s 3 1.655 s 2 0.4978 s 0.0039 from the Inverse Laplace Transform. Using numde, sym2poly and residue, see fib4.m for details, returns r 1 =0.0029262.8394-474.1929-1.0857-9.0930-0.3326211.3507 r 1 0.0029 262.8394 -474.1929 -1.0857 -9.0930 -0.3326 211.3507 and p 1 =-1.3565-0.2885-0.1667-0.1667-0.1667-0.1667-0.0100 p 1 -1.3565 -0.2885 -0.1667 -0.1667 -0.1667 -0.1667 -0.0100 You will be asked in the exercises to show that this indeed jibes with the x 1 t=211.35ⅇ-t100−0.0554t3+4.5464t2+1.085t+474.19ⅇ-t6+ⅇ-329t400262.842cosh73t16+262.836sinh73t16 x 1 t 211.35 t 100 0.0554 t 3 4.5464 t 2 1.085 t 474.19 t 6 -329 t 400 262.842 73 t 16 262.836 73 t 16 achieved in the Laplace Transform via ilaplace.

Comments, questions, feedback, criticisms?

Send feedback