On Euler's Identity

16 Dec, 2023

It has always been an interesting theme of how to introduce the Euler's Identity to highschool students. Maclaurin series is one possible choices but it is subtle to prove its convergence and hard to catch up with, not mentioning the dubious extension of complex differentiability of exponential function $e^{z}$ .

Let's start with the argument function $a r g (\cdot)$ , as this relationship for unit complex numbers $\cos θ + i \sin θ$ can be proven quite elementarily:

a r g (u v) = a r g (u) + a r g (v) .

Which means that $a r g$ function satisfies the functional equation:

f (x y) = f (x) + f (y) .

We can prove, with ease, that non-zero function $f (x)$ has to be a logarithmic function if it is defined on real numbers $ℝ$ .

Thus we would guess that the unit complex number can be written as an exponential function of its argument.

\cos θ + i \sin θ = α^{θ}

The only problem is differentiation. At this stage we know nothing about the nature of the seemingly innocent $α$ , I would pretend it being real. A second-order differentiation would yield:

\frac{d^{2}}{d θ^{2}} \cos θ + i \sin θ = \frac{d^{2}}{d θ^{2}} α^{θ},

- \cos θ - i \sin θ = α^{θ} (\ln α)^{2} .

Notice that the latter is equivalent to $- α^{θ}$ , thus we have $(\ln α)^{2} = - 1$ , and this gives us $α = e^{i}$ . Plug back this relationship results the classic

\cos θ + i \sin θ = e^{i θ} .

An important defect in this introduction would be the differentiation of $α^{θ}$ when $α$ is clearly not a real number. Considering that the same defect presents in the trivial Maclaurin approach (infinitely many times), my approach would be milder to students.

Also, this approach naturally leads to the definition of logarithm function on $ℂ$ :

L o g z : = \ln | z | + A r g (z) i .

#"hs-maths"