The formula is simple, if not straightforward:
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
We know the Maclaurin series of the functions
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 .
Exponential functions having a complex value in the exponent can also be evaluated:
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