4 ms·
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our p
by jlokier 3y ago
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix to be 1.*
No, it doesn't work in degrees.
The definition of e isn't that arbitrary.
2π is the unique period which satisfies the definition of e using derivatives and the extension of real number algebraic laws to complex numbers. This shows up as a real world physical measurement, which I describe below.
The (natural) exponential function eˣ is defined as the unique function which equals its own derivative and satisfies e⁰ = 1 (like other exponentials). The value of e comes from this.
Combine that with the definition i² = -1 and using basic rules of algebra which are observed on real numbers with exponentials and derivatives (such as (xª)ᵇ = xªᵇ) and you find the function eˣ must be periodic with period 2πi.
This comes from sin(x) and cos(x) and their derivatives. The derivative of sin(x) is cos(x), and of cos(x) it is -sin(x), but only if sin(x) and cos(x) are defined in the usual math way with period 2π.
Those sin/cos derivatives and that little negative sign are enough to make them components of the unique solution to the derivative definition of eˣ applied to a complex argument, and thereby fix its period in the complex plane and prove Euler's famous identity (without needing the Taylor expansion).
That in turn has.a more physical basis. Asin(x+B) with constants A, B are the family of functions whose second derivative equal themselves negated.
Physically, it means an object whose acceleration is proportional to its displacement from a fixed position and in the opposite direction will oscillate with a period of exactly 2π seconds, if the acceleration is -1m/s² per 1m displacement.
This setup is called a harmonic oscillator.
In this way, 2π arises (and is measurable!) from physical properties of time, force and inertia, of things moving in straight lines.
No circles required.
- nlitened 3y agoI agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for example, Euler's formula below would still hold: e^ix = cos x + i sin x And people in comments would rave about how magic number 360 is, and its magic properties were discovered by Romans two thousand years ago.
- codeflo 3y agoYou seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline. exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ... (BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't be too hard to see because it's already a Tailor series.) Purely from this you can prove that exp(ix) with real x is periodic with period 6.28... It just so happens that this number is also the circumference of the unit circle.
- quickthrower2 3y agoI had never spotted that before, each term of the series is the integral of the previous. That is pleasing! The Pi thing feels now less of a coincidence than the fact that exp is a power. That probably falls out of expanding the polynomials but it so ingrained as taken for granted that it is wonderous when you think about it.
- nlitened 3y agoHey, you're right! So Pi is special :)
- quickthrower2 3y agoI guess radians are “magic” in that sin and cos can be defined by infinite series that look nice (and feel canonical). You have to manipulate those series to get 360 or even just revolutions.
- nlitened 3y agoAs codeflo showed in a sibling comment, actually I am wrong, and Pi's special magic also becomes from Taylor's series expansion. So turns out that Pi is the real magic number rather than just our arbitrary choice!