I am currently doing some reading on using Fourier transforms to solve PDEs and I stumbled upon a property that I am not sure how to prove.
Suppose we have a heat-equation $u_t(x,t)=\alpha^2 u_{xx}(x,t)$ with $(x,t)\in \mathbb R\times [0,\infty)$. Now we can use a Fourier transform in regards to $x$, which yields
$$\mathcal F(\alpha^2 u_{xx})=-\alpha^2 \xi^2 \mathcal F(u)$$
This is obvious. But now my textbook says that
$$\mathcal F(u_t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\partial}{\partial t}u e^{-ix\xi}dx=\frac{1}{\sqrt{2\pi}}\frac{d}{dt}\int_{-\infty}^\infty ue^{-ix\xi}dx = \frac{d}{dt}\mathcal F(u)$$ and I am not really sure why this holds.
I think I have read a theorem, a long time ago, that said something about changing the order of integrals and derivatives by using Lebesgue's theorem. Is this the way to go? Or is there an easier way to argue why the above holds?
