A simple example of a complex function is the function \[f(z)=z+1+2\,\mathrm{i}\] The below dynamic figure shows the effect on points in the complex plane; the red draggable point is the original and the green point is the image under the given function \(f\). For clarity, we have also drawn the vector corresponding to the complex number \(1+2\,\mathrm{i}\) and the same vector shifted such that the starting point coincides with the red dot. It should be clear that the function trnaslates points in the complex plane via the vector that is associated with the complex number \(1+2\,\mathrm{i}\). The translation vector is here equal to \(\begin{pmatrix} 1\\ 2\end{pmatrix}\).

Translation in a plane as a complex function The complex cunction \(f(z)=z+a+b\,\mathrm{i}\) with real numbers \(a\) and \(b\) corresponds with a translation in the complex plane with translation vector \(\begin{pmatrix} a\\ b\end{pmatrix}\).

Scaling in a plane as a complex function The complex function \(f(z)=a\cdot z\) with \(a\) a real number corrsponds with scaling with respect to the origin by a factor of \(a\).

Scaling-rotation-translation in a plane as a complex function The complex function \(f(z)=(a+b\,\mathrm{i})\cdot z+c+d\,\mathrm{i}\) with real numbers \(a\), \(b\), \(c\) and \(d\) corresponds with transformation which consists of the rotation about the origin by the angle \(\mathrm{Arg}(a+b\,\mathrm{i})\) and the multiplication with respect to the origin by a factor of \(|a+b\,\mathrm{i}|\), followed by the translation with the translation vector \(\begin{pmatrix} c\\ d\end{pmatrix}\).