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