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 $z$ such that in an infinitesimally small neighborhood of a point $a$ in the complex plane, $f(\xi )=z\xi$ for an infinitesimally small vector $\xi$. The complex number $z$ is defined to be the derivative of $f$ at $a$.

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.

Examples

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