In addition to its role as a fundamental mathematical result, Euler's formula has numerous applications in physics and engineering.
A straightforward proof of Euler's formula can be had simply by equating the power series representations of the terms in the formula:
Suppose is complex. Then
which leads to the very famous Euler's identity:
Recall that we have that
Note: This means that is not a well defined (unique) quantity. To remedy this, one needs to specify a branch cut. For example, we can define the argument of to be defined for , in which case we have that . That is, this forces . Of course, different branch cut can be chosen yielding different values for .
Euler's formula allows for any complex number to be represented as , which sits on a unit circle with real and imaginary components and , respectively. Various operations (such as finding the roots of unity) can then be viewed as rotations along the unit circle.
One immediate application of Euler's formula is to extend the definition of the trigonometric functions to allow for arguments that extend the range of the functions beyond what is allowed under the real numbers.
A couple useful results to have at hand are the facts that
It follows that
Solve in the complex numbers.
We first note that if is a solution, then so is for any integer . This is because is an even function with a fundamental period of .
Hence, for any integer .
Euler’s formula also allows for the derivation of several trigonometric identities quite easily. Starting with
Equating the real and imaginary parts, respectively, yields the familiar sum and difference formulas
An important corollary of Euler's theorem is de Moivre's theorem.
De Moivre's Theorem
We have For , we have This implies that Thus, we have
De Moivre's theorem has many applications. As an example, one may wish to compute the roots of unity, or the complex solution set to the equation for integer . Notice that is always equal to for an integer, so the roots of unity must be
This process is akin to dividing the unit circle up into equally spaced wedges.
Find the cube roots of unity.
The cube roots of unity are