Complex variable function refers to a function with complex numbers as independent and dependent variables, and the related theory is complex variable function theory. Analytic function is a kind of analytic function in complex variable function. Complex variable function theory mainly studies analytic functions in complex number field, so it is usually called analytic function theory.

origin

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.

Overview of development:

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.