Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. It involves the analysis of how these functions behave under transformations and mappings in the complex plane.
A mapping in complex analysis is a transformation that takes points in the complex plane and maps them to other points in the same plane. It can be represented by a function that takes a complex number as an input and produces another complex number as an output.
The image of a mapping is determined by applying the mapping to each point in the domain. This results in a new set of points in the range, which is the image of the original set of points under the given mapping.
The image of a mapping is the set of points in the range that are produced by applying the mapping to points in the domain. The preimage, on the other hand, is the set of points in the domain that map to a given point in the range. In other words, the preimage is the inverse of the mapping.
Finding the image through a mapping allows us to understand how functions behave under transformations in the complex plane. This is essential for solving complex mathematical problems and for applications in fields such as physics, engineering, and economics.