Mapping diagram for linear complex function. I[br][br]A complex linear function [math]f(z)=a+bz[/math] where [math]a,b\in\mathbb{C}[/math] has some qualities similar to a real linear function, but when [math]b\in\mathbb{C}[/math] is a complex number, there is a distinct geometric feature: the function magnifies [math]z[/math] by |b| while rotating the result, in the plane by [math]Arg(b)[/math] and then translating that result by the vector corresponding to [math]a[/math]. This process is sometimes described as a translated [b]amplitwist[/b].[br][br]The mapping diagram can be used to understand this visually.[br]Check the box[b] Hide/show MD of function.[br][br][/b][list][*][b]Move the complex number b in the control frame on the horizontal axis to see how the magnification works. [/b][/*][/list][list][*][b]Move the complex number b off the horizontal to see how the amplitwist works. [/b][/*][/list][list][*][b]Move the complex number a in the control frame to see how the translation works. [br][/b][/*][/list]Check the box[b] Show/hide lines to see the cone or hyperbaloid of one sheet surfaces that captures some of this geometry for the circle [math]|z|=r.[/math][/b]

Mapping diagram for linear complex function. I[br][br]A complex linear function [math]f(z)=a+bz[/math] where [math]a,b\in\mathbb{C}[/math] has some qualities similar to a real linear function, but when [math]b\in\mathbb{C}[/math] is a complex number, there is a distinct geometric feature: the function magnifies [math]z[/math], by |b| while rotating the result, in the plane by [math]Arg(b)[/math] and then translating that result by the vector corresponding to [math]a[/math]. This process is sometimes described as a translated [b]amplitwist[/b].[br][br]The mapping diagram can be used to understand this visually.[br]Check the box[b] Hide/show MD of function.[br][br][/b][list][*][b]Move the complex number b in the control frame on the horizontal axis to see how the magnification works. [/b][/*][/list][list][*][b]Move the complex number b off the horizontal to see how the amplitwist works. [/b][/*][/list][list][*][b]Move the complex number a in the control frame to see how the translation works. [br][/b][/*][/list]Check the box[b] Show/hide lines to see the cone or hyperbaloid of one sheet surfaces that captures some of this geometry for the circle [math]|z|=r.[/math][/b]

Use the slider to change the radius of the circle, r, in the domain for the mapping diagram.