[mcv]

Is there an easy explanation of why these are true? I already struggle grasping e^i, and I completely don't understand what e to an irrational power even means, let alone why it would be -1. Why the circle circumference ratio has anything to do with this is completely beyond me.

[larschdk]

3Blue1Brown: https://youtu.be/v0YEaeIClKY?si=BQ2W64DWIuhHoUuX

[photochemsyn]

1. e^x, sin(x) and cos(x) can each be expanded out into an infinite sequence of fractions (Maclaurin series), each fraction of the form x^n / factorial(n). This relies on differential calculus, Taylor series, theory of limits, convergence etc.

2. The e series fractions contain all the integers in the numerator (x^0, x^1, x^2, etc), while the sin series has only odd integers and the cosine series has only even integers. Also, e series terms are all additive while the trig functions series alternate adding and subtracting each successive fraction.

3. Introducing complex numbers (i = square root of negative one), we can generate the series for e^ix, which can be shown to be equal to sin(x) + i * cos(x). Note that introducing i into the e series means we generate a negative term for the even fractions in the e series (squaring i gives us -1), which is why i is so necessary here.

4. Solving e^ix for x = pi, using sin(x) + icos(x), we get -1.

Mathologer:

https://www.youtube.com/watch?v=-dhHrg-KbJ0

and

https://www.youtube.com/watch?v=DoAbA6rXrwA

As far as why an exponential function like e^x should have anything fundamental linking it to trigonometric functions like sin(x) and cos(x), it is rather strange.

[seanhunter]

So that's the part I explain in my note. So if you combine the two explanations together I think we have the whole picture. Yours fills in the gap in my explanation.

[seanhunter]

So I haven't done complex analysis yet which I think you need to get the whole thing but I can get you some of the way there with basic trig.

If you take a unit circle and construct a radius to some point (x,y), if you drop a perpendicular line down to the x-axis, it's easy to see that the length of that perpendicular line is y and the distance you've gone across the x axis is x. So you have a right angled triangle where the hypotenuse is 1 (it's a unit circle) and the other two sides are x and y. Now consider the angle at the origin and call that theta.[1] You can do basic trig to show that the coordinates of your (x,y) point are (cos theta, sin theta), because sin is opposite (y) over hypoteneuse (1) and cos is adjacent (x) over hypotenuse. Ok cool. So x = cos theta and y = sin theta. If you measure in radians, then the angle of a full revolution is 2 * pi radians and the angle of a half revolution (180 degrees in other words) is pi. Now consider the point when you have gone around the unit circle 180o, Its coordinates are x=-1 and y=0. Remember this point - we'll come back to it in a minute.

Now imagine instead of your unit circle being just any old circle it's in the complex plane. This means that the x axis is the real part of some complex number and the y axis is the imaginary part. We now know that the coordinates of points on this circle are (cos theta, sin theta), but if you have a complex number z= a+bi, these correspond to a and b. So z = cos theta + i sin theta. Here's the bit where my current mathematical ability runs out of gas and you're just going to have to trust Euler, who showed that cos theta + i sin theta = e^(i theta).

Now remember our point from before where theta = 180 degrees? What was the angle in radians? It was pi. So e^(i pi) = -1 (because the real part of the number is the x coordinate, -1 and the imaginary part, the y coordinate is zero).

[1] Here's a diagram I made which will get you up to here https://www.geogebra.org/calculator/btz38m3c. My note about the trig of unit circle I made while studing is here https://publish.obsidian.md/uncarved/3+Resources/Public/Unit...