stylistic:tone, linguistic:paraphrasing, information:classification
Other applications

In differential equations, the function e 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.

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 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 quaternions, there is a sphere of imaginary units. For any point r on this sphere, and x a real number, Euler's formula applies: exp x r = cos x + r sin x, {\displaystyle \exp xr=\cos x+r\sin x,} and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.