mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the amplitwist is a concept created by

Tristan Needham Tristan Needham is a British mathematician and professor of mathematics at the University of San Francisco. Education, career and publications Tristan is the son of social anthropologist Rodney Needham of Oxford, England. He attended the Dragon ...

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

a

Uses

The concept of an amplitwist is used primarily in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

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^

. Since the derivative of

f(z)

3z^2

, we can say that for an infinitesimal vector

\gamma

e^

f(\gamma)=3(e^)^2\gamma = 3e^{i\frac{\pi}{2\gamma

References

Functions and mappings Complex analysis