Though it’s probably the most commonly used definition for complex exponentiation, e^i pi=-1 is not an axiom— it is derived from the basic rules of integer arithmetic expanded in scope until they make a closed system.

And from this, we can derive all of the trigonometric functions using only algebra and no geometry.

(See http://www.feynmanlectures.caltech.edu/I_22.html , particularly section 22-5)

> I think e^i pi = -1 is drastically overrated. That is typically the definition of exponentiating to a complex power, that e^ix = sin x + i cos x.

Yes, but it's the definition of exponentiating to a complex power because it is easily proven by reference to the real Taylor series of those functions. The complex definition is motivated by the real result, not -- as you imply -- the other way around.

(Also, you have it backwards; e^ix is cos x + i sin x.)

You can easily derive e^ix = i cos x + sin x.

Just look at the Taylor series for e^x and compare with the Taylor series for sin x + cos x. You'd see that they are basically the same, just that the trigonometric functions alternate their signs every 2 steps. How do we alternate the signs in e^x? We just add a factor of i, that way we get alternate signs every two steps but get an additional i on every odd step. Odd steps are sin, so just add i to sin and we are done: e^ix = i sin x + cos x.

Now we also see that you remembered the wrong formula, it the sine part is imaginary since it corresponds to the odd factors.

I have no problems with this route to show that e^ix = cos(x) + i sin(x) but to do it properly does require showing that this is the unique analytic continuation of the exponential function.

The alternative method is to note that exp() is defined as the unique function satisfying exp(0) = 1 and dexp(ax)/dx = a exp(ax). So exp(ix) describes a point moving orthogonal to it's current position, which means it's a circle, and since it's speed is proportional to it's distance from the origin it is moving at a constant speed. So e^ix is just a point x radions around the unit circle.

(it's the other way round, e^(ix) = cos(x) + i sin(x))

Addition is commutative.

Not sure whether that was meant as a joke or not, but the difference is which term is real and which is imaginary, not the order of the terms...