Complex variable functions are functions defined on the complex plane. Holomorphic functions are an important part of complex variable functions. However, there are many names for holomorphic functions. What is the relationship between these names? For example, for the same object, someone sees this feature and calls it this name, and someone sees another feature and calls it another name. The most commonly used name for a holomorphic function, some people call it an analytic function, why is it called an analytic function? This is because the holomorphic function in any area can be expanded into a power series at any point in the area, and the power series is an expression, which is an analytic expression, so people who see this point call the holomorphic function an analytic function . Because power series are widely used and analytic formula is a commonly used mathematical tool, but the concept of analytic function has not been found in other mathematics disciplines, so analytic function has become synonymous with holomorphic function.
A differentiable function is a name for a holomorphic function. This is because looking at a complex function from a differential point of view, a complex function is differentiable if it is differentiable at one point. Points are analytic points, which are holomorphic points, and more importantly, a holomorphic function can calculate derivatives of any order in the region, that is, it can be differentiated at any order. When people see this characteristic and property of holomorphic functions, they Holomorphic functions are called differentiable functions, but the differentiable functions here are different from the differentiable functions in mathematical analysis. The differentiable functions here refer to differentiable complex functions, and they are arbitrarily differentiable. is named from the aspect of differentiability of holomorphic functions.
The regular function is also the name of the holomorphic function. The regular function originally refers to the function that can be expanded into a power series in advanced mathematics. It is called the regular function. This definition is extended to the complex variable function, and the regular function is the complex variable function. The holomorphic function in is also named from the characteristics of the analytic function.
The shape-preserving function has a name. This is a holomorphic function from the perspective of the change of the geometric figure. The independent variable of the complex variable function is a complex number, and the function value is also a complex number. Such a four-dimensional figure cannot be made, so people Thinking of using one plane as the plane where the independent variable is located, and the other plane as the plane of the function value, the complex variable function can study the corresponding relationship between the graphics on these two planes, and in this corresponding relationship, the holomorphic function as a mapping has The shape-preserving feature, so the holomorphic function is also called a shape-preserving map or a shape-preserving transformation or a shape-preserving function.
Source: Public Account - Complex Analysis Mathematics