Amplitwist

In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually.

Definition

The amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number ${\displaystyle z}$ such that in an infinitesimally small neighborhood of a point ${\displaystyle a}$ in the complex plane, ${\displaystyle f(\xi )=z\xi }$ for an infinitesimally small vector ${\displaystyle \xi }$. The complex number ${\displaystyle z}$ is defined to be the derivative of ${\displaystyle f}$ at ${\displaystyle a}$.^[1]

Uses

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]

Examples

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 ${\displaystyle e^{i{\frac {\pi }{4}}}}$, ${\displaystyle f(\gamma )=3(e^{i{\frac {\pi }{4}}})^{2}\gamma =3e^{i{\frac {\pi }{2}}}\gamma }$.