Euler Formula

The formula which connects the complex exponential with trigonometric motion was discovered by Leonhard Euler around 1740, and it is called Euler's formula in his honor.

EULER'S FORMULA

$$\begin{equation*}e^{i\theta} = \cos \theta + i \sin \theta.\end{equation*}$$

It says that the complex number $e^{i\theta}$ moves on the unit circle, with real part $\cos\theta$ and imaginary part $\sin\theta$.

When $\theta=\pi$, Euler's formula gives the identity

$$\begin{equation*}e^{i\pi} + 1 = 0,\end{equation*}$$

which combines the constants $e$, $i$, $\pi$, $1$, and $0$ in a single equation.

Applications

Trigonometry

Euler's formula gives a clean way to derive trigonometric identities. The key idea is that multiplication of complex exponentials turns addition of angles into multiplication:

$$\begin{equation*}e^{i(\theta+\varphi)}=e^{i\theta}e^{i\varphi}.\end{equation*}$$

Using Euler's formula on both sides, we get

$$\begin{equation*}\cos(\theta+\varphi)+i\sin(\theta+\varphi)=(\cos\theta+i\sin\theta)(\cos\varphi+i\sin\varphi).\end{equation*}$$

Expanding the product gives

$$\begin{equation*}\underbrace{(\cos\theta\cos\varphi-\sin\theta\sin\varphi)}_{\cos(\theta+\varphi)}+i\underbrace{(\sin\theta\cos\varphi+\cos\theta\sin\varphi)}_{\sin(\theta+\varphi)}.\end{equation*}$$

Now compare real and imaginary parts. The real parts give

$$\begin{equation*}\cos(\theta+\varphi)=\cos\theta\cos\varphi-\sin\theta\sin\varphi,\end{equation*}$$

and the imaginary parts give

$$\begin{equation*}\sin(\theta+\varphi)=\sin\theta\cos\varphi+\cos\theta\sin\varphi.\end{equation*}$$

Thus one complex equation contains two familiar trigonometric identities at once.

References

• [Needham2023]Needham, Tristan. \textit{Visual Complex Analysis}. 25th Anniversary Edition. Oxford University Press, 2023. First edition published in 1997.