Euler's Formula

Introduction

This formula is named after Leonhard Euler and is an important formula in the analysis of complex numbers. This formula was first discovered by Roger Cotes in 1714, even though the formula is named after Euler. It shows the relationship between the complex exponential and the trigonometric functions sine and cosine.

Euler's Formula

\,\mathrm{e}^{j x} = (\cos x + j\sin x )\,

Proof of Euler's Formula

This can be proved in one of several ways, using calculus (derivatives), differential equations, or the Taylor series, which is used here.

The functions e^x, cos x and sin x of the (real) variable x can be expressed using their Taylor expansions around zero:

\begin{align} e^x &{}= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \\ \cos x &{}= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ \sin x &{}= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \end{align}

For complex z we define each of these functions by the above series, replacing the real variable x with the complex variable z. This is possible because the radius of convergence of each series is infinite. We then find that

\begin{align} e^{iz} &{}= 1 + iz + \frac{(iz)^2}{2!} + \frac{(iz)^3}{3!} + \frac{(iz)^4}{4!} + \frac{(iz)^5}{5!} + \frac{(iz)^6}{6!} + \frac{(iz)^7}{7!} + \frac{(iz)^8}{8!} + \cdots \\ &{}= 1 + iz - \frac{z^2}{2!} - \frac{iz^3}{3!} + \frac{z^4}{4!} + \frac{iz^5}{5!} - \frac{z^6}{6!} - \frac{iz^7}{7!} + \frac{z^8}{8!} + \cdots \\ &{}= \left( 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots \right) + i\left( z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots \right) \\ &{}= \cos z + i\sin z \end{align}

The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.