Inversion mapping of z is a mathematical concept in complex analysis where the mapping of points in the complex plane is inverted or flipped across a circle or line. This results in a new mapping where the image of points outside the circle/line are mapped inside and vice versa.
The purpose of inversion mapping is to provide an alternative way of visualizing the complex plane and its points. It also helps in understanding the behavior of analytic functions and their transformations.
Inversion mapping is a special case of conformal mapping, where the mapped image maintains the angles and preserves the orientation of curves. Inversion mapping is conformal only at the points where the circle/line of inversion intersects with the original curve.
Inversion mapping has various applications in physics and engineering, such as in the study of electromagnetic fields, fluid dynamics, and heat transfer. It is also used in computer graphics and image processing for geometric transformations.
One example of inversion mapping is the stereographic projection, where the complex plane is projected onto a sphere by inverting the north pole. Another example is the Poincaré disk model, where the complex plane is mapped onto the unit disk by inverting its boundary circle.