Английская Википедия:Amplitwist

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Шаблон:Short description 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>z</math> such that in an infinitesimally small neighborhood of a point <math>a</math> in the complex plane, <math>f(\xi) = z \xi</math> for an infinitesimally small vector <math>\xi</math>. The complex number <math>z</math> is defined to be the derivative of <math>f</math> at <math>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>f(z) = z^3</math>. Consider the derivative of the function at the point <math>e^{i\frac{\pi}{4}}</math>. Since the derivative of <math>f(z)</math> is <math>3z^2</math>, we can say that for an infinitesimal vector <math>\gamma</math> at <math>e^{i\frac{\pi}{4}}</math>, <math>f(\gamma)=3(e^{i\frac{\pi}{4}})^2\gamma = 3e^{i\frac{\pi}{2}}\gamma</math>.