Since Cauchy, the theory of complex variable function has a history of 150 years, and it has become an important part of mathematics with its perfect theory and exquisite skills. It promotes the development of some disciplines and is often used as a powerful tool in practical problems. With the development of physics disciplines such as fluid mechanics, the length of stable surface and aviation mechanics, many branches of mathematics have also applied its theory. The theory of complex variable function has penetrated into calculus equations, number theory and other disciplines, which has had a great influence on their development. For example, Russian Rukovski used the theory of complex variable function to solve the structural problems of aircraft wings when designing aircraft, and he also made contributions to solving problems in fluid mechanics and aviation mechanics by using the theory of complex variable function. The theory of complex variable function has become a very important part of mathematics with its perfect theory and exquisite skills. The theory of complex variable function promotes the development of many disciplines and is also a powerful tool to solve some practical problems. The theory and method of complex variable function are widely used in mathematics, natural science and engineering technology, and are powerful tools to solve plane problems such as fluid mechanics, electromagnetism, heat and elasticity theory.
Complex function can be applied to geographic information system. Because of the calculation requirements of complex function and the analysis of spatial function in GIS, the application of complex function has also penetrated into this field, and its calculation ability of complex function makes its application in GIS indispensable. The operating objects of GIS are spatial data and attribute data, that is, geographical entities with three-dimensional elements such as points, surfaces and bodies. The most fundamental feature of spatial data is to encode each data according to unified geographical coordinates, and realize its positioning, qualitative and quantitative description, which is a technical difficulty. Riemann surface theory in complex variable function is used to solve this problem. Complex variable function studies multivalued functions, and Riemann surface theory is the main tool to study multivalued functions. A surface composed of many layers is called a Riemannian surface. Using this surface, the concepts of single-valued branch and multi-valued function branch can be expressed and explained intuitively in geometry. For a multivalued function, if its Riemannian surface can be made, then the function becomes a single-valued function on the Riemannian surface.
The application of complex variable function mainly includes two aspects: one is the application in physics; On the other hand, it is applied in the field of mathematics.
The application of 1. complex function in electrostatic field physics
Application of complex function in electrostatic field;
In the study of electromagnetic field, "the scalar potential of electrostatic field" comes into contact with the application of complex variable function in electrostatic field problems. That is, if a system is a two-dimensional electrostatic field system, the distribution of its field and source is only related to x and y. Because in the two-dimensional passive region, the electrostatic potential satisfies the two-dimensional Laplace equation, that is
V (x.y)=(x.y) (x，y)
ax2 oy2
We find that the point at this time is a harmonic function. Through the study of complex variable function, we already know that the real part and imaginary part of analytic function are both harmonic functions and a pair of harmonic functions. Therefore, we can use the mathematical tool of complex variable function to solve the two-dimensional electrostatic field problem.
Therefore, the concept of complex potential is introduced into electromagnetic field. If f(2)=(X, y)+iv (,),
ou(x，y)0u(x，y) ( 1)
Arkansas
0'v(x. y) v(.x.y' (2)
ax' ay