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 Leffler, French mathematician Poincare and Adama. A lot of research work has been done, which has opened up a broader research field of complex variable function theory and contributed to the development of this discipline.

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.

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.

The content of complex variable function theory

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. The recent research on 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 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. At present, 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.