Some Proofs of Euler's Equation
The usual proof that you will meet concerning this rather wonderful little equation is the one where you show that the power series expansions of the three separate functions in the equation are consistent with the equation. I hate this proof, it leaves me cold!
I much prefer to take a few risks by making some outrageous assumptions (notably that complex numbers and calculus are able to mix freely), so these are far from rigorous proofs, but I like them anyway. The first is my favourite, some neat tricks and a quick and easy solution. The second is more involved, but it is interesting anyway.
The great thing about this final relationship and the reason it is often called ‘beautiful’ is that it contains all five fundamental constants of the universe (that’s e, 1, 0, i and pi) and details a simple relationship containing only those five and nothing else.