Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive understanding of euler's apparently magical formula: $$e^{ \pm i\theta } = \cos \theta \pm i\sin \theta$$
I've obviously seen the taylor series/differential equation based proofs, and perhaps I'm just going to have to accept that it's not possible to have an intuition on what it means to raise a number to an imaginary power. I obviously realise that the formula implies that an exponential with a variable imaginary part can be visualised as a complex function going around in a unit circle about the origin of the complex plane. But WHY is this? And why is e so special that it moves at just a fast enough rate so that the argument of the exponential is equal to the arc length of the path made by the locus (i.e. the angle in radians we've moved around the circle)? Is there any way anyone out there 'understand' this?
Thankyou!
Answer
If I recall from reading Analysis of the Infinite (very nice book, at least Volume $1$ is), Euler got it from looking at
$$\left(1+\frac{i}{\infty}\right)^{\infty}$$
whose expansion is easy to find using the Binomial Theorem with exponent $\infty$.
There is a nice supposed quote from Euler, which can be paraphrased as "Sometimes my pencil is smarter than I am." He freely accepted the results of his calculations. But of course he was Euler.
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