It is easy enough to make any given function complex by fixing the domain to some complex region and "seeing what happens". Let us break this down even further:[br][br]Let [math]z[/math] be a complex number of the form:[br] [math]z=x+yi[/math][i], [/i]where [math]x,y\in\mathbb{R}[/math][br]We can view [math]x[/math] as the real component and [math]y[/math] as the imaginary component of the complex number [math]z[/math].[br][br]Let u,v take values for x,y and return another real value such that:[br] [img]https://latex.codecogs.com/gif.latex?u%2Cv%3A%5Cmathbb%7BR%7D%5E2%5Crightarrow%20%5Cmathbb%7BR%7D[/img][br]where f can then be written with the form:[br] [math]f\left(z\right)=f\left(x,y\right)=u\left(x,y\right)+v\left(x,y\right)i[/math][br]Now we have a general equation for Complex Functions and can start talking about differentiation of [math]f[/math].[br][br][br]