Complex function

Writing \(w=f(z)\) with \(z=x+iy\) amounts to giving two real functions of two variables, \(u(x,y)\) and \(v(x,y)\). For instance, \(f(z)=z^2\) has \(u=x^2-y^2\) and \(v=2xy\), and \(f(z)=\bar z\) has \(u=x\), \(v=-y\). This decomposition is what later links complex differentiability with the Cauchy-Riemann equations.

The domain of \(f\) is the set \(D\) where the rule makes sense; unless stated otherwise, the largest possible one is taken. Thus \(1/(z-i)\) is defined on \(\mathbb{C}\setminus\{i\}\), which is a domain in the sense of plane topology. The image is the set of values \(f(z)\) with \(z\in D\).

A graph would need four dimensions, so a function is visualized as a transformation: draw two planes, \(z\) and \(w\), and see what lines, circles or regions become. The function \(f(z)=iz\) rotates the plane by \(90^\circ\); \(f(z)=z^2\) doubles angles at the origin and squares moduli, so it sends \(|z|=2\) to \(|w|=4\).

Sums, products and quotients of complex functions are defined pointwise, and composition works as in the real case. Frequent mistakes: confusing \(|z|^2\) (real) with \(z^2\) (complex) and forgetting the cross term when expanding \((x+iy)^2\). The next step is to give meaning to \(f(z)\to L\) as \(z\to z_0\).