A few weeks back I asked a question which lead to Euler's formula being brought up. I don't have the mathematical background to fully appreciate it's purported mathematical beauty.
Just yesterday I wondered about the graph of the function $(-1)^x$ and after some exploration and research was again lead back to Euler's formula.
I'm wondering if someone can provide an explanation of Euler's formula sufficient to help me appreciate it in the context of periodic functions.
So what functions solve the DE $y" = -y$? That is, what functions can you differentiate twice, take the negative, and get the function back?
They are $\sin(x), \cos(x), e^{ix}$. However, because the original question was a second order DE, there is only two solutions, not three. That must mean $e^{ix}$ can be written as $A\sin(x) + B\cos(x)$. Solving for those constants gets you Euler's formula.