Euler's Formula¶

Theorem¶

For any $z\in\mathbb{C}$$\mathbb{C}$, we have $\mathrm{e}^{\mathrm{i}z}=\cos z+\mathrm{i}\sin z.$

Interpretation¶

Euler's formula connects the complex exponential with trigonometry. For a real angle $\theta$ it says that $\mathrm{e}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$ is the point of modulus one and angle $\theta$ in the polar coordinates of the complex plane.

The point $\mathrm{e}^{\mathrm{i}\theta}$ lies on the unit circle at the angle $\theta$ from the positive real axis, and the dashed lines meet the axes at its real part $\cos\theta$ and its imaginary part $\sin\theta$. As $\theta$ grows the point runs anticlockwise round the circle: $\theta=0,\pi/2,\pi,3\pi/2$ give the marked points $1$, $\mathrm{i}$, $-1$ and $-\mathrm{i}$. So $|\mathrm{e}^{\mathrm{i}\theta}|=1$ for every real $\theta$: multiplying by $\mathrm{e}^{\mathrm{i}\theta}$ is a rotation by $\theta$, not a stretch.

The reason lies in the series defining the complex exponential and the complex sine and cosine. Substitute $\mathrm{i}z$ into the exponential series. The powers of $\mathrm{i}$ repeat in the cycle $1,\mathrm{i},-1,-\mathrm{i}$, so the even terms contain no $\mathrm{i}$ and alternate in sign, which gives the cosine series, while the odd terms all contain a factor $\mathrm{i}$ and give $\mathrm{i}$ times the sine series.

With the formula, the polar form becomes $z=r\mathrm{e}^{\mathrm{i}\varphi}$$r{\mathrm{e}}^{\mathrm{i}\phi }$, and De Moivre's formula becomes $(\mathrm{e}^{\mathrm{i}\varphi})^n=\mathrm{e}^{\mathrm{i}n\varphi}$${\mathrm{e}}^{\mathrm{i}n\phi }$; the angle $\pi$ gives Euler's identity. The unit-circle reading is for real angles only. The formula holds for every complex $z$, but then $\cos z$ and $\sin z$ need not be real and $\mathrm{e}^{\mathrm{i}z}$ need not lie on the unit circle: for $z=\mathrm{i}$$\mathrm{i}$, $\mathrm{e}^{\mathrm{i}\cdot\mathrm{i}}=\mathrm{e}^{-1}\approx0.37$$0.37$.

Key Results¶

• We have $|\mathrm{e}^z| = \mathrm{e}^{\operatorname{Re}(z)}$ and $\arg(\mathrm{e}^z) = \operatorname{Im}(z) + 2\pi k .$
• We have $\cos\theta = \dfrac{\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta}}{2}$ and $\sin\theta = \dfrac{\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}}{2\mathrm{i}}.$

Examples¶

• When $z=\pi$$\pi$, we get the well-known Euler's identity.

Prerequisites¶

• Complex Exponential Function
• Complex Trigonometric Functions
• Complex Number
• Complex Plane Polar Coordinate System

Related Concepts¶

• De Moivre's Formula
• Complex Exponentiation
• Euler's Number

Sources¶

• Math II: Linear Algebra · Week 1

Symbols: complex numbers, element of, phi, pi, theta

Last updated: 20 October 2026 03:02