Component functions u and v

The real and imaginary parts of $f(z)$ , written as two-variable real functions $u(x,y)$ and $v(x,y)$ .

Writing $z=x+iy$ , any complex function $f(z)=u+iv$ splits into its component functions $u(x,y)$ and $v(x,y)$ : ordinary real-valued functions of the two real variables $x$ and $y$ , giving the real part and imaginary part of $f(z)$ respectively. Trading one complex function for this pair of real ones is what lets calculus tools built for real functions of two variables -- partial derivatives, in particular -- get applied to complex functions.

Examples

For $f(z)=e^z$ , the component functions are $u(x,y)=e^x\cos y$ and $v(x,y)=e^x\sin y$ .
Knowing $u$ and $v$ separately at every point of a domain is exactly equivalent to knowing $f$ there.