mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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, best known to the public for his books ''Visual Complex Analysis'', and ''Visual Differential Geometry and Forms''. Education, career and p ...

in the book ''Visual Complex Analysis'' (1997) to represent the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

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