The other relationship between these values is given by Euler's formula:

Euler's formula involves a complex power of a positive real number and this always has a defined value.

This article is about Euler's formula in complex analysis.

The compact case arises through Euler's formula in the complex plane.

The complex number forms in the definitions above derive from Euler's formula.

After some manipulation utilizing Euler's formula, the following result is obtained.

The proof is an easy consequence of Euler's formula.

In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

A case that involves complex roots can be solved with the aid of Euler's formula.

This is another version of Euler's formula, but with a general bivector in three dimensions.

One can also produce them algebraically using Euler's formula.

Another simple "proof" can be given using Euler's formula known from complex analysis:

He develops their theory and finds, among other things, the higher dimensional version of Euler's formula.

Next, use the inverse Euler's formula to obtain complex sinusoids.

Furthermore, Euler's formula was probably known before the life of Euler.

Euler's formula is ubiquitous in mathematics, physics, and engineering.

The fundamental solutions and can be further rewritten using Euler's formula:

These formulas come from the definition, Euler's formula and elementary trigonometric identities.

This can be shown using Euler's formula.

He provides the first proof of what is now known as Euler's formula and constructs the logarithmic spiral.

Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface.

It can be noted through Euler's formula and its derivatives that and .

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

There are many proofs of Euler's formula.

This is the quaternionic analogue of Euler's formula.