Editing Euler's formula (section)

===Other applications===
{{see also|Complex number#Applications}}

In [[differential equation]]s, the function {{math|''e<sup>ix</sup>''}} is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the [[eigenfunction]] of the operation of [[differentiation (mathematics)|differentiation]].

In [[electrical engineering]], [[signal processing]], and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see [[Fourier analysis]]), and these are more conveniently expressed as the sum of exponential functions with [[imaginary number|imaginary]] exponents, using Euler's formula. Also, [[phasor analysis]] of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

In the [[four-dimensional space]] of [[quaternion]]s, there is a [[sphere]] of [[imaginary unit]]s. For any point {{mvar|r}} on this sphere, and {{mvar|x}} a real number, Euler's formula applies:
<math display="block">\exp xr = \cos x + r \sin x,</math>
and the element is called a [[versor]] in quaternions. The set of all versors forms a [[3-sphere]] in the 4-space.