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Fourier integral theorem

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Verify that exp(-bx) where b is a positive constant satisfied the conditions of the Fourier integral theorem given in our book (see below).

    2. Relevant equations

    N/A

    3. The attempt at a solution

    The theorem says under what conditions the Fourier series applies. The conditions are that f must be absolutely integerable for x > 0 and piece wise continuous on every bounded interval on it. Also, f(x) at each point of discontinuity of f must be the mean value of the one-sided limits f(x+) and f(x-). f(x) represents the Fourier integral, and f represents the original function. The theorem says that if these conditions hold, you can write f as the Fourier series f(x).

    I am supposed to verify the conditions mentioned above for e^(-bx) where b is a positive constant.

    My question is how is exp(-bx) piecewise smooth on every bounded interval of it? It is not a piece wise function.
     
  2. jcsd
  3. Apr 14, 2009 #2
    It is "piecewise", it only has one piece, and that piece is continuous on all of R.
     
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