Euler's equation

Complex numbers can be expressed either as pairs of numbers (real and imaginary parts) or in polar form, as an amplitude and phase (angle from real axis in the complex plane). Euler's equation gives a particularly compact and useful polar representation.

Euler's equation

First proof: differential equations

Two functions are equivalent if they have the same value at one point, and satisfy the same differential equation.

Proof 1 of Euler's equation

Second proof: series expansion

Two functions are equivalent if they satisfy the same series expansion.

Proof 2 of Euler's equation

© 1999-2005 Randy J Read, University of Cambridge. All rights reserved.

Last updated: 7 June, 2005