A complex function is a function $f$ from a subset $A$ of $\mathbb{C}$ to $\mathbb{C}$ .

For every $z = x+iy\in A\,\,\,(x,\,y \in \mathbb{R})$ the complex value $f(z)$ can be split into its real and imaginary parts $u$ and $v$ , respectively, which can be considered as real functions of two real variables: $$f(z) = u(x,\,y)+iv(x,\,y)$$ The functions $u$ and $v$ are called the real part and the imaginary part of the complex function $f$ , respectively. Following are the notations for $u$ and $v$ that are used most commonly (the parentheses around $f(z)$ may be omitted):

$$u(x,\,y) = \mbox{Re}\left(f(z)\right) = \Re\left(f(z)\right)$$ $$v(x,\,y) = \mbox{Im}\left(f(z)\right) = \Im\left(f(z)\right)$$

The branch of mathematics concerning differentiable complex functions is called function theory or complex analysis.