Amplitwist

Short description: Concept used to represent a derivative

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

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