/kloc-in the 9th century, the theory of complex variable functions gradually became an independent branch of mathematics, and Cauchy did basic work for it.

Complex variable function and complex power series

More than half of the course "Analysis" discusses complex numbers and elementary complex variable functions, which shows that Cauchy has long regarded the establishment of complex variable function theory as an important project in analysis. He introduced complex numbers ("imaginary expressions") in a formal way, defined their basic operations and obtained the properties of these operations. He defined the complex infinitesimal and the continuity of complex variable function according to the actual situation.

Cauchy's paper on definite integral in 18 14 is his first step to establish the theory of complex variable function. This paper gives the so-called cauchy-riemann equations. This paper discusses the order change of double integrals, puts forward the concept of principal value integral when the integrand has infinite discontinuous points, and calculates many generalized integrals.

Cauchy's paper on definite integral with integral limited to imaginary number in 1825 is a masterpiece. In this paper, Cauchy integral theorem is put forward as the basis of simple complex variable function theory. Cauchy himself proved this theorem by variational method, and proved that the idea of continuous deformation of the middle curve can be said to be the germination of the concept of "homotopy". This paper also discussed the calculation of generalized integral when the integrand has first-order and m-order poles.

The word residue in residue calculus first appeared in a paper written by Cauchy in 1826. He believes that residue calculus has become "a new calculation method similar to calculus" and can be applied to a large number of problems.

Establishment of complex variable function theory