Complex Analysis

The complex exponential function exp(z), sin(z), cos(z) are defined as power series, used to prove properties analogous to the real case, with new ideas such as Euler's formula exp(z) = cos(z) + i.sin(z). Defining e = exp(1) gives exp(z) = e^z. For z= x + iy, e^z= e^(x + iy) = e^x.e^(iy) = e^x(cos(y) + i sin(y)). Real properties of the exponential and the trigonometric functions are used to build the complex generalisations, linking symbolic properties to visual dynamic complex representations. These include the periodicity of exp, sine and cosine and introduction of other complex trigonometric and hyperbolic functions.