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e (mathematical constant) - Wikipedia, the free encyclopedia:

The exponential function e^x may be written as a Taylor series

Because this series keeps many important properties for e^x even when x is complex, it is commonly used to extend the definition of e^x to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

which holds for all x. The special case with x = π is known as Euler's identity:

Consequently,

from which it follows that, in the principal branch of the logarithm,

Furthermore, using the laws for exponentiation,

which is de Moivre's formula.

The case,

is commonly referred to as Cis(x).