Comment by ndriscoll

You could argue this is just moving the memorization to meta-facts, but I found all throughout school that if you understand some slightly higher level key thing, memorization at the level you're supposed to be working in becomes at best a slight shortcut for some things. You can derive it all on the fly.

Sort of like how most of the trigonometric identities that kids are made to memorize fall out immediately from e^iθ = cosθ+isinθ (could be taken as the definitions of cos,sin), e^ae^b=e^(a+b) (a fact they knew before learning trig), and a little bit of basic algebraic fiddling.

Or like how inverse Fourier transforms are just the obvious extension of the idea behind writing a 2-d vector as a sum of its x and y projections. If you get the 2d thing, accept that it works the exact same in n-d (including n infinite), accept integrals are just generalized sums, and functions are vectors, and I guess remember that e^iwt are the basis you want, you can reason through what the formula must be immediately.