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Interchanging integration and partial derivative

  1. Dec 26, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]f\in L_{loc}^1(\mathbb{R}_+)[/itex].
    Need show that for Re[itex](z)>\sigma_f[/itex] (abscissa of absolute convergence) we have $$\mathcal{L}[tf(t)](z)=-\frac{d}{dz}\mathcal{L}(z)$$where [itex]\mathcal{L}[/itex] denotes Laplace transform.

    3. The attempt at a solution
    The proof comes down to whether $$\int_0^\infty\frac{\partial}{\partial z}\left(e^{-zt}f(t)\right)dt=\frac{d}{dz}\int_0^\infty e^{-zt}f(t)dt$$holds.
    All the theory on switching integration and derivative I could find requires the integration interval to be finite and/or f to be continuous which is not really the case.
    Any ideas welcome.
     
  2. jcsd
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