Math people. Do you have an intuitive *geometric* reason why interlacing the power series of sin x and cos x gives the series for eˣ?
Do you have a way that you *visualize* growth whose rate is exactly its present value coming out of the unit circle as sine and cosine do?
I know the equations, and the proofs and they aren't convoluted and probably "enough" but I hope someone can share their mental images. Even if they aren't fully formed.
There is a nice attempt in Visual complex analysis, by Tristan Needham.
He first encourages you to see exp(x) as (1 + x/n)**n for n-> infinity. Interest paid out continuously, you know.
That has a nice geometric equivalent for complex exp(x + iy). Namely, a stack of n triangles with height y/n. As n goes to infinity, that stack is a spiral, or a circle if the real part is zero
Edit: and the circle then maps to cos and sin
Later he has introduced the complex derivative as 'amplitwist', where the imaginary part is the twist of the mapping.
Then you can define exp(z) as the complex function whose complex derivative equals itself, and that becomes a circle for exp(iy)
@Zamfr @futurebird came here to recommend this book! I had forgotten the exact analogies but I found them very interesting back in the day.
