Definition of the sine 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 connecting the exponential, the sine, and the cosine functions:
Changing to , the Euler formula becomes:
Taking the difference of the preceding formulas and dividing by 2ⅈ gives the following result:
Here are two graphics showing the real and imaginary parts of the sine function over the complex plane.