## Euler's Equation

In this section, we consider the important complex function,

Substituting with its definition, :

The part is easy to understand, since it is the ordinary exponential function with a real argument, but what about the second exponential on the right hand side of the equation? What does it mean for an exponent to be imaginary? To find out, we can expand the term about :

Grouping the real and imaginary parts,

The real part is composed of the even terms of the expansion, while the imaginary part is composed of the odd terms. It is easy to verify that

and

Thus,

This is known as

*Euler's formula.*If we set , we find that

or

This is known as

*Euler's identity*. It is significant because it relates the fundamental numbers, , , , , and using multiplication, exponentiation, addition, and equality. Euler's formula also gives us another way to write the imaginary unit:

And this allows us to express the th root of as

Returning to the original expression,