Euler's Formula

Theorem
Let $z \in \C$ be a complex number.

Then:
 * $e^{i z} = \cos z + i \sin z$

where:
 * $e^{i z}$ denotes the complex exponential function
 * $\cos z$ denotes the complex cosine function
 * $\sin z$ denotes complex sine function
 * $i$ denotes the imaginary unit.

Real Domain
This result is often presented and proved separately for arguments in the real domain:

Also known as
Euler's formula in this and its corollary form are also found referred to as Euler's identities, but this term is also used for the specific example:
 * $e^{i \pi} + 1 = 0$

It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.

Also see

 * Euler's Identity
 * Sum of Hyperbolic Sine and Cosine equals Exponential