Complex function

A rule $w=f(z)$ assigning a complex number to each complex number in some domain.

A complex function $f$ assigns a complex number $w=f(z)$ to every complex number $z$ in some set (its domain of definition). Writing $z=x+iy$ and $w=u+iv$ , every complex function splits into a pair of real, two-variable functions $u(x,y)$ and $v(x,y)$ -- the real and imaginary parts of the output -- so a single complex function secretly packages two ordinary real functions traveling together.

Examples

$f(z)=z^2$ sends $2+i$ to $(2+i)^2=3+4i$ .
For $f(z)=z^2$ , writing $z=x+iy$ gives $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$ .