# Homework Help: Interchanging integration and partial derivative

1. Dec 26, 2012

### TaPaKaH

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