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.
The amplitwist associated with a given function is its derivative in the complex plane.
More formally, it is a complex number
such that in an infinitesimally small neighborhood of a point
in the complex plane,
for an infinitesimally small vector
The complex number
is defined to be the derivative of
[1] 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.
[1][2] Define the function
Consider the derivative of the function at the point
Since the derivative of
, we can say that for an infinitesimal vector