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.
