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Showing an integral doesn't converge

  1. May 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that the following function is not square integrable, i.e. that it is not continuous.

    [tex]

    \int_{-\infty}^{\infty} \left ( e^{ikx} \right )^{2}dx

    [/tex]

    2. Relevant equations

    See above. Also:

    [tex]

    \int \left ( e^{ikx} \right )^{2}dx = -\frac{ie^{2ikx}}{2k}

    [/tex]

    3. The attempt at a solution

    [tex]

    =\lim_{A \rightarrow -\infty}\int_{A}^{C} e^{2ikx}dx+\lim_{B \rightarrow \infty}\int_{C}^{B} e^{2ikx}dx

    [/tex]

    How do I go from there? What would I choose for C? Can it be anything?
     
    Last edited: May 17, 2010
  2. jcsd
  3. May 17, 2010 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Not being square integrable is not the same as not being continuous. f(x) = 1 is not square integrable on (-oo, oo) but is obviously continuous.

    Yes, you can use anything for C. And if either integral diverges the whole thing does.
     
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