When can I commute the 4-gradient and the "space-time" integral?

[tannhaus]

TL;DR

I do not know how to procceed in a situation where I have a 4-gradient and a space-time integral.

Let's say I have the following situation

$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$

Would I be able to commute the integral and the partial derivative? If so, why is that? In the same line of thought, in the situation I'm able to commute, would the result of this be

$$I = \int k_{\alpha}e^{k_{\mu}x^{\mu}}\;d^4k$$

Thanks in advance!

 

[renormalize]

Would I be able to commute the integral and the partial derivative? If so, why is that?

Since:$$e^{k_{\mu}x^{\mu}}=e^{k_{0}x^{0}}e^{k_{1}x^{1}}e^{k_{2}x^{2}}e^{k_{3}x^{3}}$$you only need to consider the behavior of a single integral of the form $\int e^{kx}dk$. So compare:$$\frac{d}{dx}\left\{ \int e^{kx}dk\right\} =\frac{d}{dx}\left\{ \frac{e^{kx}}{x}\right\} =\frac{e^{kx}\left(kx-1\right)}{x^{2}}$$to:$$\int\left\{ \frac{d}{dx}\left(e^{kx}\right)\right\} dk=\int\left\{ ke^{kx}\right\} dk=\frac{e^{kx}\left(kx-1\right)}{x^{2}}$$These are equal so it's clearly OK to commute differentiation and integration in your situation.

 

[haushofer]

See also Leibniz's integral rule:

https://en.wikipedia.org/wiki/Leibniz_integral_rule

or the lecture notes

https://kconrad.math.uconn.edu/blurbs/analysis/diffunderint.pdf

wheren also counterexamples are discussed.