Amplitwist

The concept of an amplitwist is used primarily in complex analysis to offer a way of visualizing the derivative of a complex-valued function as a local amplification and twist of vectors at a point in the complex plane.^[1][2]

Define the function ${\displaystyle f(z)=z^{3}}$. Consider the derivative of the function at the point ${\displaystyle e^{i{\frac {\pi }{4}}}}$. Since the derivative of ${\displaystyle f(z)}$ is ${\displaystyle 3z^{2}}$, we can say that for an infinitesimal vector ${\displaystyle \gamma }$ at ${\displaystyle e^{i{\frac {\pi }{4}}}}$, ${\displaystyle f(\gamma )=3(e^{i{\frac {\pi }{4}}})^{2}\gamma =3e^{i{\frac {\pi }{2}}}\gamma }$.