Amplitwist

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