brief introduction
The concept of complex number originated from finding the roots of equations. When finding the roots of quadratic and cubic algebraic equations, the square root of negative numbers appeared. For a long time, people couldn't understand this figure. However, with the development of mathematics, the importance of such numbers is increasingly apparent.
The theory of complex variable function came into being in18th century. 1774, Euler considered two equations derived from the integration of complex variables in one of his papers. Before him, French mathematician D'Alembert had obtained them in his paper on fluid mechanics. Therefore, people later mentioned these two equations and called them "D'Alembert-Euler equations". In the19th century, when Cauchy and Riemann studied fluid mechanics, they studied the above two equations in more detail, so they were also called Cauchy-Riemann conditions.
The comprehensive development of complex variable function theory was in the19th century. Just as the direct expansion of calculus ruled the mathematics in the18th century, the new branch of complex variable function also ruled the mathematics in the19th century. Mathematicians at that time recognized that the theory of complex variable function was the richest branch of mathematics, which was called the mathematical enjoyment of this century. Some people praise it as one of the most harmonious theories in abstract science.
Euler and D'Alembert did the earliest work for the establishment of the theory of complex variable function, and French Laplace later studied the integration of complex variable function. They are all pioneers in establishing this subject. Later, German mathematicians Cauchy and Riemann made peace and did a lot of basic work for the development of this discipline. At the beginning of the 20th century, the theory of complex variable function has made great progress. Students of Wilstrass, Swedish mathematician Levreux, French mathematicians Poincare and Adama all did a lot of research work, which opened up a broader research field of complex variable function theory and made contributions to the development of this discipline.
The theory of complex variable function is widely used not only in other disciplines, but also in many branches of mathematics. It has been deeply involved in differential equation, integral equation, probability theory and number theory, and has had a great influence on their development.
content
Complex variable function theory mainly includes single-valued analytic function theory, Riemann surface theory, geometric function theory, residue theory, generalized analytic function and so on.
If a function has a unique fixed value when its variable takes a certain value, then the solution of the function is called a single-valued analytic function, and a polynomial is such a function.
Complex functions also study multivalued functions, and Riemann surface theory is the main tool to study multivalued functions. A surface composed of many layers put together is called a Riemannian surface. Using this surface, the concepts of single-valued bifurcation and multi-valued function bifurcation can be expressed and explained intuitively by geometry. For a multivalued function, if its Riemannian surface can be made, then the function becomes a single-valued function on the Riemannian surface. Riemann surface theory is a bridge between complex variable function domain and geometry, which enables us to relate the analytical properties of relatively abstruse functions with geometry. At present, the study of Riemannian surfaces has a great influence on topology, another branch of mathematics, and gradually tends to discuss its topological properties.
In the theory of complex variable function, the content of explaining and solving problems by geometric methods is generally called geometric function theory, and complex variable function can provide geometric explanation for its properties through * * * shape mapping theory. The images realized by analytic functions whose derivatives are not zero everywhere are all * * * images, which is also called conformal transformation. * * * images have been widely used in fluid mechanics, aerodynamics, elasticity theory, electrostatic field, circuit theory and so on. Residue theory is an important theory in complex variable function theory. Remainder is also called residue, and its definition is complicated. It is more convenient to calculate the integral of complex variable function by residue theory than by line integral. The calculation of definite integral of real variable function can be transformed into the integral of complex variable function along closed-loop curve, and then transformed into the calculation of residue of integrand function on isolated singularity inside closed-loop curve by using the basic theorem of residue. When the singularity is the pole, the calculation is more concise.
Some conditions of single-valued analytic function are modified and supplemented appropriately to meet the needs of practical research work. The analytic function of this change is called generalized analytic function. The change of geometric figure expressed by generalized analytic function is called quasi-conformal transformation. Some basic properties of analytic functions can also be applied to generalized analytic functions with a little change. Generalized analytic functions are widely used, not only in the study of fluid mechanics, but also in solid mechanics departments such as thin shell theory. Therefore, the theory in this field has developed very rapidly in recent years.
Since Cauchy, the theory of complex variable function has a history of 170 years. It has become an important part of mathematics with perfect theory and exquisite skills. It promotes the development of some disciplines and is often used as a powerful tool in practical problems. Its basic content has become a compulsory course for many science and engineering majors. There are still many topics to be studied in the theory of complex variable function, so it will continue to develop and get more applications.
The main research object of complex variable function is analytic function, including single-valued function, multi-valued function and geometric theory. In the long historical process, with the efforts of many scholars, the theory of complex variable function has made great progress and formed some special research fields.
The two basic types of single-valued functions are integer functions and meromorphic functions, which are the development of polynomials and rational functions respectively. Wilstrass extended the factorization theorem of polynomials to whole functions, and G. Mitta-Levler extended the theorem of decomposing rational functions into partial fractions to meromorphic functions. Piccard, female. -J.-) éBorer and others further found that the values of whole functions and polynomials are very similar. On this basis, 1925 R. Nevanlinna established a modern theory of meromorphic function value distribution, which had an important influence on the development of function theory. It is also closely related to other fields of complex variable function theory. For example, 1973 A. Bernstein introduced T * function with the idea of real variable function, which played a significant role in the study of deficiency problem of value distribution theory, minimum modulus problem of whole function and univalent function.
The research on multivalued functions is mainly around Riemannian surfaces and univalence problems. 19 13 (C.H.) H. Weil gave the definition of abstract Riemannian surface for the first time in his classic book The Concept of Riemannian Surface, which is the embryonic form of manifold and the basic concept of modern mathematics. The study of Riemannian surface not only forms a perfect theory, but also provides a simple and clear model for the study of important branches of modern mathematics such as algebraic geometry, automorphic function, complex manifold and algebraic number theory.
In the application of complex variable function, * * shape mapping plays an important role. Your Excellency zhukovsky's research on the flow around the wing through the shape diagram is a famous example. In practical applications, it is often necessary to construct mapping functions by means of approximate methods. There is a lot of research work in this field. Of course, sometimes you don't need to know the specific mapping function, just apply its geometric properties. This promotes the development of geometric theory of complex variable functions.
The study of univalent function is an important part of geometric theory of complex variable function, especially the conjecture that univalent analytic function has the shape of formula (4) in the unit circle put forward by L. Bieber Bach in 19 16, which has attracted the attention of many scholars. In recent 70 years, there has been a lot of research work on Bieber Bach's conjecture, but it was not until 1984 that Blanqui completely confirmed this conjecture. In the proof, the work of Lebada-Millin, the parameter representation of C. Lowner and the results about Jacobian polynomials are mainly applied.
Cauchy-riemann equations revealed the relationship between analytic function and elliptic partial differential equation. Since the 1950s, L. Bos, иу. Vequa et al. considered the general elliptic partial differential equation and introduced the concept of generalized analytic function. The mapping determined by analytic function is a * * * shape mapping, which maps an infinitesimal circle into an infinitesimal circle; The generalized analytic function determines the quasi-* * shape mapping, which maps an infinitesimal circle into an infinitesimal ellipse. L.V. Ahlfors and м α Lavrentev laid the foundation of quasi-* * shape mapping theory.
Although the analytic function has good properties in the region, when the independent variable z approaches the boundary, the change of the function is often very complicated. The research in this field has formed a special field called boundary properties of analytic functions. Classical results include the normal graph theorem, η η Luzin and ии. Privalov has also made a systematic study in this respect. The emergence of the concept of aggregation has further triggered further research.
The theory of complex variable function involves a wide range of applications, and many complex calculations are solved by it. For example, there are many different stable plane fields in physics. The so-called field is a region, each point corresponds to a physical quantity, and their calculation is solved by complex variable function. 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 the problems of fluid mechanics and aviation mechanics with the theory of complex variable function.