Proving Laplace Transform is Analytic

I was trying to prove that the Laplace Transform is analytic and to do so I tried to evaluate its derivative which is (this needs to be proven) $$F'(s)=-\int_{0}^{\infty}tf(t)e^{-st}dt$$ where f(t) is a complex valued function and s is complex. If you split the real and Imaginary part of the integrand and suppose that you can differentiate under integral sign I see that the Cauchy Riemann equations are satisfied then the function is analytic. What I am having trouble proving is the differentiation under integral sign. The books I consulted just differentiate without justifying the passages and most of them use the following hypotheses: 1)f(t)=0 for all t<0 2)there exists a real number $$a$$ such that $$\int_{0}^{\infty}|f(t)e^{-at}|dt$$ exists and is finite, $$g(t)=f(t)e^{-at}$$ is such that $$g(t)=1/2(g(t_+)+g(t_-))$$ is piecewise differentiable on every closed interval and has domain $$(-\infty,\infty)$$. So basically from these conditions what I'd like to prove is that for all s in the complex domain where the laplace transform is defined then $$\int_{0}^{\infty}tf(t)e^{-st}dt$$ converges (continuity follows from piecewise differentiability of g(t) and product of continuous functions is continuous)

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