The exponential function for complex variables. Power series approach. [br][math]e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots[/math][br]The red vectors represent each term in the serie. [br][br]Notice also that if [math]z=x+iy[/math], then [math]e^z=e^{x+iy}=e^{x}e^{iy}[/math]. Thus, the green vector represents [math]e^x[/math], and the angle is the imaginary part of [math]z[/math], that is, [math]y[/math].