Re: The Euler Equation: What does "e" have to do with sine & cosine?


MichaelDMcDonnell@xxxxxxxxx wrote:
First, thank you to all the smart people who responded to my last
Euler Equation post...

I'm trying to understand this equation from an intuitive perspective.
For example, what does the number "e" have to do with sine and cosine?
We have
e = 1/0! + 1/1! + 1/2! + 1/3! + ...
and for real x we have
cos x = x^0/0! - x^2/2! + x^4/4! - ...
sin x = x^1/1! - x^3/3! + x^5/5! - ...
Now if we just put x=i and hope that nothing goes wrong, we get
cos i - i sin i = e.

(One moment, please, while I LOL at my own joke.)

But seriously, if multiplication by -1 is a half-turn of the plane,
then it is reasonable to model multiplication by i as a quarter-turn,
since i is to be a square root of -1. From there, rotation by an
arbitrary angle turns out to be multiplication by x+iy with xx+yy=1.
Indeed to provide a fully rigorous _definition_ of sine and cosine, it
is quite feasible to bring the complex exponential. Bourbaki, in
particular, does it that way.

LH

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