One of the most important identities in all of mathematics, Euler's formula relates complex numbers, the trigonometric functions, and exponentiation with Euler's number as a base.

The formula is simple, if not straightforward:

Alternatively:

When Euler's formula is evaluated at

, it yields the simpler, but equally astonishing Euler's identity.

As a consequence of Euler's formula, the sine and cosine functions can be represented as

Derivation

We know the Maclaurin series of the functions

Therefore

Trigonometry

The formula permits the extension of the trigonometric functions to complex-valued domains and ranges. In other words, it is possible to find complex, unrestricted values for which

and

In addition, it permits the determination of complex-valued inputs for values outside of the normal range of the trigonometric functions. In other words, if is complex, then it is possible that

even though the domain of the cosine function is normally restricted to the real interval .

Exponentials

Exponential functions having a complex value in the exponent can also be evaluated:

Applications

Euler's formula is used extensively in complex analysis. It is also used often in differential equations, as Euler's number being raised a complex variable appears fairly often.

An interesting corollary of Euler's formula is that can be found and is entirely real.

However, does not have one singular representation, as the function is multivalued and depends on which branch is chosen. The general form for any integer is