Intuition for Euler's formula by using the limit $\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$?

Question

My goal is to give good intuition as to why the formula $$e^{i\theta}=\cos\theta+i\sin\theta$$

Is correct. I don't need to be formal or rigorous.

One good way of doing this is by looking at the taylor expansion of $e^x$ for real values of $x$, substituting $x=i\theta$ and getting the taylor expansions for $\cos$ and $\sin$.

What I want: To give a similar intuitive explanation without using taylor expansions.

What I do have: The equality

$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$

So, I can ask myself what happens with this formula when I substitute $x=i\theta$. But I'm not sure what happens now - can we really proceed from this starting point to get an intuition for Euler's formula?

Yes, @robjohn posted a detailed answer related to this somewhere. I can’t quite find the post right now though — FShrike, Dec 08 '22 at 11:21
For an intuitive understanding, try looking at $e^{i\theta}e^{i\phi} =e^{i(\theta+\phi)}$ and $(\cos\theta+i\sin\theta)(\cos\phi+i\sin\phi)$ compared to $\cos(\theta+\phi)+i\sin(\theta+\phi)$ to see that they behave as you might expect. — Henry, Dec 08 '22 at 11:24
$f(\theta)=\cos\theta+i\sin\theta$ has the properties $f(0) = 1$ , $f(x)f(y) = f(x+y)$ and $f'(x) = i f(x)$ . The only solution is $f(x) = e^{ix}$ — user3257842, Dec 08 '22 at 11:32
A simple proof of the limit in question for all complex values of $x$ is given in this answer of mine. — Paramanand Singh, Dec 22 '22 at 04:50

Meni Rosenfeld · Answer 1 · 2022-12-17T19:12:54.067

You could denote $\mathrm{cis}\theta=\cos\theta+i\sin\theta$, and show (with induction and trigonometric angle addition formulas) that $\mathrm{cis}(n\theta) = (\mathrm{cis}\theta)^n$. (Showing first that $\mathrm{cis}(\theta+\phi)$ = $\mathrm{cis}\theta\cdot\mathrm{cis}\phi$ may or may not be easier.)

Then use the fact that for small $\theta$, $\sin\theta\approx\theta$ and $\cos\theta\approx 1$ to find $e^{i\theta}\approx\left(1+\frac{i\theta}{n}\right)^n\approx\left(\mathrm{cis}\left(\frac{\theta}{n}\right)\right)^n=\mathrm{cis}\theta=\cos\theta + i\sin\theta$, where the approximation becomes exact as $n\to\infty$.

score -1 · Answer 2 · answered Dec 08 '22 at 11:29

-1

In order to have a better understanding of Euler’s formula, you should picture yourself the two-dimensional complex plane. In fact, the right side of the equation describes the circular motion along the outside of the unit unit circle using sine and cosine. That is, if we set $\theta = \pi$, then that means that we travel $\pi$-units along the unit circle. We are then half way around the circle, i.e. we’re at $-1 = e^{i\pi} = \cos(\pi) + i\sin(\pi)$.

answered Dec 08 '22 at 11:29

Dylan Schmidt

69

It is a good intuition and I like to use it, but I need to relate it to one of the standard definitions of $e^x$; in this case, the limit-based one. That's the content of my question. – Gadi A Dec 10 '22 at 06:59
When taking into account the given definition of exp(x), it can be shown that exp($i\theta$) will have magnitude 1 and angle x for $\lim_{n \to \infty}$, which is the same as the expression on the right side of the equation. – Dylan Schmidt Dec 10 '22 at 12:56

Intuition for Euler's formula by using the limit $\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$?

2 Answers2