Further evidence that
e^{ix} = cos(x) + i sin(x)
is natural is that it fits in nicely with power series representations. If you define cos(x), sin(x), and e^x by their power series centered at 0 then it's straightforward to see that substituting ix into the power series for e^x yields the sum of the power series for cos(x) and i*sin(x) (as long as you accept that theorems about absolute convergence and rearranging terms extend to the complex numbers).
Re:The exponential function is a miracle
by tonyarkles on Friday September 13, 2019 @09:56AM
(#20962167)
The lecture in my Calculus IV course where we walked through that derivation is literally my only memorable math lecture in university. It just so beautifully ties so many different concepts together!
df/dx = f
The solutions are e^x + const; requiring f(0) = 1 gives you the choice const = 0.
Another way to define it is as the infinite series
e^x = \sum_{i=0}^inf x^i/i!
It is pretty easy to show this definition is equivalent to the previous one. A third way is as
lim_{n -> inf} (1 + x/n)^n
This definition comes from the intuition that e^x represents the limit of continually compounding interest. As before, it is pretty easy to show that it is equivalent to the previous two.
In any case, all these definitions extend directly to complex values of x. The fact that
e^{ix} = cos(x) + i sin(x)
holds (and hence Euler's identity holds) is a consequence of a more natural definition, not the typical definition.