Definition of the cosine function for a complex argument

In the complex ‐plane, the function is defined using the exponential function in the points and through the formula:

The key role in this definition of belongs to the famous Euler formula that connects the exponential, the sine, and the cosine functions:

Changing to , the Euler formula can be converted into the following modification:

Adding the preceding formulas gives the following result:

Here are two graphics showing the real and imaginary parts of the cosine function over the complex plane.