Amplitwist Explained

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

such that in an infinitesimally small neighborhood of a point

in the complex plane,

f(\xi)=z\xi

for an infinitesimally small vector

\xi

. The complex number

is defined to be the derivative of

.^[1]

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.^[2]

Examples

Define the function

f(z)=z³

. Consider the derivative of the function at the point

i	\pi
	4

. Since the derivative of

f(z)

3z²

, we can say that for an infinitesimal vector

\gamma

i	\pi
	4

i	\pi
	4

f(\gamma)=3(e

)^2\gamma=

i	\pi
	2

\gamma

References

Book: Tristan., Needham. Visual complex analysis. 1997. Clarendon Press. 0198534477. Oxford. 36523806.
Soto-Johnson . Hortensia . Hortensia Soto . Hancock . Brent . February 2019 . 10.1080/10511970.2018.1477889 . 5 . . 421–440 . Research to Practice: Developing the Amplitwist Concept . 29.