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Chapter 5e^(iπ) in 3.14 minutes, using dynamics

The Real Case

One way to think about the function ete^t is to ask what properties it has. Probably the most important one, from some points of view the defining property, is that it’s a function which is equal to its own derivative. Together with the added condition that inputting 00 returns 11, it’s the only function with this property.

You can illustrate what that means with a physical model: Think of ete^t as your position on a number line as a function of time. The condition e0=1e^0 = 1 means you start at 11. The equation above is saying that your velocity, the derivative of position, is always equal to your position. In other words, the farther away from 0 you are, the faster you move, with the very specific constraint that your velocity vector is perpetually identical to your position vector.

So even before knowing how to compute ete^t exactly, this ability to associate each position with the velocity you must have at that position paints a very strong intuitive picture of how the function must grow. You know you’ll be accelerating, at an accelerating rate, with an all-around feeling of things getting out of hand quickly.

If we add a constant to this exponent, like e2te^{2t}, the chain rule tells us the derivative of our function is now 22 times the function itself.

So at every point on the number line, rather than attaching a vector corresponding to the number itself, first double the magnitude, then attach it. Moving so that your position is always e2te^{2t} is the same thing as moving in such a way that your velocity is always twice your position. The implication of that 22 is that our runaway growth feels all the more out of control.

If that constant was negative, say -0.5, then your velocity vector is always -0.5 times your position vector, meaning you flip it around 180-degrees, and scale its length by a half.

Moving in such a way that your velocity always matches this flipped and squished copy of the position vector, you’d go the other direction, slowing down in exponential decay towards 0.

The complex case

What about if the constant was ii, the imaginary unit? If your position was always eite^{it}, how would you move as that time, tt, ticks forward?

Well, assuming there’s any way to make sense out of eite^{it}, and assuming that derivatives still work the way we’d expect when extending to complex numbers, the derivative of your position eite^{it} would now always be ii times itself. Multiplying by ii has the effect of rotating numbers 90-degrees, and as you might expect, things only make sense here if we start thinking beyond the number line and in the complex plane.

Geometrically, which of the following describes the point i(a+bi)i \cdot (a + bi) on the complex plane?

So what does eite^{it} mean?

So even before you know how to compute eite^{it}, you know that for any position this might give for some value of t, the velocity at that time will be a 90-degree rotation of that position.

Each blue arrow above pointing outward from the origin represents an example value of $e^{it}$, thought of as a position vector. The corresponding green arrow at its tip shows what the velocity, equal to the position rotated 90 degrees counterclockwise, would have to be for that particular position.

Each blue arrow above pointing outward from the origin represents an example value of eite^{it}, thought of as a position vector. The corresponding green arrow at its tip shows what the velocity, equal to the position rotated 90 degrees counterclockwise, would have to be for that particular position.

Drawing this for all possible positions you might come across, we get a vector field, where, usually with vector fields we shrink things down to avoid clutter.

At time t=0t=0, eite^{it} will be 1. There’s only one trajectory starting from that position where your velocity is always matching the vector it’s passing through, a 90-degree rotation of position. It’s when you go around the unit circle at a speed of 1 unit per second.

So after π\pi seconds, you’ve traced a distance of π\pi around; eiπ=1e^{i\pi} = -1.

After τ=2π\tau = 2\pi seconds, you’ve gone full circle; eiτ=1e^{i\tau} = 1.

And more generally, eite^{it} equals a number tt radians [1] around this circle.

Nevertheless, something might still feel immoral about putting an imaginary number up in that exponent. And you’d be right to question that! For values of xx which are not real numbers, what we write as exe^x has very little to do with repeated multiplication, and its relation to the number ee feels frankly more incidental than definitional. To dig in a little bit deeper, the next lesson covers more general exponents, with a focus on matrices.

Footnotes

[1] - radian(s), SI unit of angle, 1 radian is equal to an angle at the center of a circle where the arc length of the circular sector is equal to the radius of the circle

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