A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

UnkleRhaukus

  • 4 years ago

Show that \[\mathcal{L} \{{1 \over x}f(x)\} =\int _s^∞F(s)ds\]

  • This Question is Closed
  1. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    The RHS is equal to \[ \int_s^\infty \int_0^\infty f(x) e^{-sx} dx \ ds \] Now swap the order of integration and you'll see it's not hard to show that this must be equal to \[ \int_0^\infty \frac{1}{x}f(x) e^{-sx} dx \]

  2. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    how do we swap order of integration again

  3. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    As ever, draw a diagram first and see what the region is. Then figure out how the limits change. In this case, it's pretty straight forward. Try it first and tell if you're stuck.

  4. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    figure out how the limits change when you change the order of integration that is.

  5. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    They may not change at all; you'll need to convince yourself one way or another.

  6. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1328840642309:dw|

  7. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Look at the region in x,s-space.

  8. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    what's a bit confusing here is that the s of limit of integration wrt to s is actually not the same variable. Which is to say, it would have been better if the RHS had been written as \[ \int_s^\infty F(s') \ ds' \]

  9. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    So look at the region in x,s'-space. It's very regular.

  10. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so x,s' space yeah

  11. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Got it?

  12. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is this the right region s' from s to ∞ x form 0 to ∞|dw:1328841005870:dw|

  13. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Yes, hence when you change the order of integration, do the limits change?

  14. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    well now x gos from 0 to ∞ and s' goes from s to ∞. the limits have not changed

  15. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok i think i have it

  16. UnkleRhaukus
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I got there thank you James

  17. JamesJ
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    It's a nice result, and what you might hypothesize, given the Laplace transform of xf(x)

  18. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.