Let's suppose that $$\frac{d}{dx}\int_a^b f(x,t)\,dt=\int_a^b\frac{\partial}{\partial x}f(x,t)\,dt$$ for all $a,b\in\mathbb{R}$.

Can we extend this result to:

$$\frac{d}{dx}\int_{-\infty}^\infty f(x,t)\,dt=\int_{-\infty}^\infty\frac{\partial}{\partial x}f(x,t)\,dt$$?

Thanks for any help.

Reason for me asking this is because there is a nice result that says differentiation under the integral is ok, provided $f$, and $f_x$ are continuous, and the domain of integration must be compact. This is much easier than finding a dominating function in case we want to use Lebesgue's Dominated Convergence Theorem.

Eventually, I am hoping to use this to solve my other question: How do we justify differentiating under the integral for Poisson Integral

  • 1
    $\begingroup$ Under suitable conditions on $\frac{\partial f}{\partial x}$, we can extend. But those suitable conditions typically amount to an invocation of the dominated convergence theorem. (For very nice $f$, you can easily get the result with more elementary tools, but few things are so nice.) If there's something not clear to you in my answer at the other question, why don't you comment and ask for clarification? I'd be happy to elaborate. $\endgroup$ – Daniel Fischer Nov 1 '16 at 13:42

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