Independent variable and dependent variable of complex functions are both complex numbers, and may be separated into real (denoted by $$\Re$$) and imaginary parts (denoted by $$\Im$$) : $$z = x +iy$$ and $$w = f(z) =u(x,y)+iv(x,y)$$ where $$x, y \in \mathbb{R}$$, $$u(x,y)$$ and $$v(x,y)$$ being real-valued functions.
A fundamental result of complex analysis is the Cauchy-Riemann equations, which give the conditions for a function to have a complex derivative, which is a complex generalization of the (real) derivative. If the complex derivative is defined at each point in an open connected subset of $$\mathbb C$$, the function is called analytic, or holomorphic. If the complex derivative is defined at each point in $$\mathbb C$$, the function is called entire.