Taking the partial derivative of an integral

Question

Can I simply take the integral of this function with respect to $t$ by bringing the differential operator under the summation?

$$u(x,t)=\int_{-\infty}^{\infty} \frac{\phi(\zeta)}{t^{1/2}}\exp\left(-\frac{(x-\zeta)^2}{4t}\right)d\zeta$$

I don't think it requires an application of Leibniz.

Answer 1

I'd consider using Leibniz's rule to solve your problem. A simplification of it says you can do the following $\frac{\partial}{\partial t} \int f(\zeta,t)d\zeta = \int \frac{\partial}{\partial t} f(\zeta,t)d\zeta$. That seems to me what you want to do. You can read about it at Leibniz Integral Rule.