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Need a solution for the following problem

  1. Jan 27, 2007 #1
    The question is attached. Thanks a lot
     

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  2. jcsd
  3. Jan 27, 2007 #2

    arildno

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    Expressed as an integral, what is F(0) equal to?
     
  4. Jan 27, 2007 #3
    F(0) = Fo where Fo can be any constant, but we have to specify it.
     
  5. Jan 27, 2007 #4
    not specify but determine actually my bad
     
  6. Jan 27, 2007 #5

    arildno

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    I SAID:
    Expressed as an integral, WHAT IS F(0)?
     
  7. Jan 27, 2007 #6
    see attached
     

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  8. Jan 27, 2007 #7

    arildno

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    Indeed!
    And what is the exact value of that integral?
     
  9. Jan 27, 2007 #8
    - e of positive infinity??
     
  10. Jan 27, 2007 #9

    arildno

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    Okay, so you are unfamiliar with the famous result:
    [tex]\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}[/tex]
     
  11. Jan 27, 2007 #10
    yea, never seen this before
     
  12. Jan 27, 2007 #11

    arildno

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    Okay, so now you know the value of F(0)! :smile:
     
  13. Jan 27, 2007 #12
    So what is the answer?
    Care to walk me through?
     
  14. Jan 28, 2007 #13
    So what do we do with this?
    Sorry, im not understanding.
     
  15. Jan 28, 2007 #14

    HallsofIvy

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    The problem SAID "Write a differential equation for F(x):
    [tex]\frac{dF}{d\omega}+ h(\omega)F[/tex]
    with F(0)= F0 where F0 is the constant you have to determine explicitly."

    Okay, you now know that F0= [itex]\sqrt{\pi}[/itex].

    By the way, I notice that the "differential equation" you give (I have copied it exactly above) is not an equation! There is no equal sign. I imagine it is actually either
    [tex]\frac{dF}{d\omega}= h(\omega)F[/tex]
    or
    [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex]

    They differ, of course, only in the sign of h.
     
  16. Jan 28, 2007 #15

    Oh so we just have to solve the ODE [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex] with initial condition F0= [itex]\sqrt{\pi}[/itex]...

    is it that straight forward? Am I not understanding something?
     
  17. Jan 28, 2007 #16
    but how do you solve a question like this, with 3 variables and a complex number? even if we use the method of solving linear differential equations
     
  18. Jan 29, 2007 #17

    HallsofIvy

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    Apparently you are having trouble reading the problem- which I just quoted.

    The problem does NOT ask you to solve any differential equation. It asks you to FIND an equation of that form (essentially find the function [itex]h(\omega)[/itex] so that F(x), as given in integral form, satisfies that equation.

    What happens if you differentiate the integral defining F?
     
  19. Jan 29, 2007 #18
    I got F(w) = root( pi) when solving the diff. equation

    So is this the answer to the integral?
     
  20. Jan 30, 2007 #19

    HallsofIvy

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    This is becoming very frustrating. Is there any point in responding if you don't read the replies?? I just said, the problem does not ask you to solve a differential equation, it asks you to find a differential equation for F! And I don't know what you mean by "the answer to this integral". Integrals don't have answers, questions do. What is the question?

    You are told that F is defined by
    [tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex]
    What do you get if you differentiate that equation with respect to [itex]\omega[/itex]? (In this case it is legitimate to simplydifferentiate inside the integral.)

    By the way, [tex]e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}[/tex]
     
  21. Jan 30, 2007 #20
    I applogize, it's not every day I come accross questions like this...this is very challenging.

    I got confuzed between "solving the differential equation" and "finding the differenatial equation of F" -- I don't know the difference. :uhh:

    The derivative of [tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex] with respect to [itex]\omega[/itex] is [tex]F(\omega)= \int_{-\infty}^{\infty}-xe^{-\omega x}e^{-x^2}dx[/tex]
     
    Last edited: Jan 30, 2007
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