What is the history and significance of Euler's formula?

[Greg Bernhardt]

Definition/Summary

Euler's formula, $e^{ix}\ =\ \cos x\ +\ i \sin x$, enables the trigonometric functions to be defined without reference to geometry.

Equations

$$e^{ix}\ =\ \cos x\ +\ i \sin x$$

and so cos and sin may be defined:

$$\cos x\ =\ \frac{1}{2}\left(e^{ix}\ +\ e^{-ix}\right)$$ and $$\sin x\ =\ \frac{1}{2i}\left(e^{ix}\ -\ e^{-ix}\right)$$

or:

$$\cos x\ =\ 1\ -\ \frac{x^2}{2} +\ \frac{x^4}{24} -\ \frac{x^6}{720}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n}}{(2n)!}$$

$$\sin x\ =\ x\ -\ \frac{x^3}{6} +\ \frac{x^5}{120} -\ \frac{x^7}{5040}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n+1}}{(2n+1)!}$$

Extended explanation

Proof of Euler's formula, starting from the trignonometric definitions of cos and sin:

Using the chain rule:

$$\frac{d}{dx}\left(e^{-ix}\,(cosx\ +\ i sinx)\right)$$

$$=\ e^{ix}\,(-i cosx\ +\ sinx\ -\ sinx\ +\ i cosx)$$

$$=\ 0$$

and so $e^{-ix}\,(cosx\ +\ i sinx)$ is a constant. Setting x = 0 we find that this constant must be 1.

and so $$cosx\ +\ i sinx\ =\ e^{ix}$$

History:

Euler's formula was discovered by Cotes.

de Moivre's formula, $(cosx\ +\ i sinx)^n$ = $cos(nx)\ +\ i sin(nx)$, is an obvious consequence of Euler's formula, but was discovered earlier.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

Euler's formula was discovered by Cotes.

Wikipedia says something different:

Euler's formula appeared for the first time in 1748 in Leonhard Euler's two-volume introductio in analysin infinitorum, first under the premise that the angle is a real number. However, this limitation soon proved superfluous, because Euler's formula applies equally to all real and complex arguments.

Anyway, $e^{i\pi}+1=0$ is considered one of the most beautiful formulas in mathematics.

There are plenty of articles out there which deal with the subject, e.g. https://sites.math.washington.edu/~marshall/math_307/complexnos.pdf