I just started reading Partial Differencial Equations. What you said is the material on Integral/Fourier Transforms.
I did not learn this yet but I did find it in my book.
Theorem: Given the heat equation with no boundary conditions i.e. it needs to be satisfied on with (some well-behaved function). Then, the solution can be written as the Fourier Integral:
Now in your problem if:
In this case the solution is given by:
Which is the Gaussian Curve, and it is very similar to how you written it.