Intuitively it seems that to get a function back that was integrated, you would take the derivative. Instead, with the Fourier Transform we take an area under a curve of a modified function, and to get the function back we again take the area under a modified resulting curve.
I am trying to more intuitively understand the concept of the FT and inverse FT. I know there is a reason this double integration works, and it seems like it has to do with the fact that the resulting function, taken one way or the other, ends up being a function of another variable that you imposed on it with e^(+/-)jwt and the integral removes the original independent variable from the equation. Maybe I need to spend more time thinking about what it means to integrate a variable out.
I'm just curious if anyone has an explanation for how the FT and inverse FT fit together that makes the functions a little more intuitive. (Sorry if the question is a little open-ended seeming, but I think it's important in mathematics to intuitively understand equations where you can. At least for me, it's always helped me a lot with gaining confidence and using the equation faster.)