? At that time, Cauchy did not give a proof, and Riemann was the first to give a proof under the condition of strengthening. With the additional condition "dw/dz is continuous in the simply connected region B", it can be proved by Green's formula. Gu Late was the first to give a complete theoretical proof, so in memory of these two mathematicians who completely guessed and proved, Cauchy Theorem is now often called Cauchy-Gusa Theorem. Relevant proof can refer to the textbook or the following papers:/link? URL = BH 1ef-CVA 5 odci 2y 1 V6 mde-uiin 2oo _ ri 6 anrvhgoypizrsyqvi 4 EC _ dugrtnfjpzlo 9 ypzs 6 lf 4 pcsmiakunj 3 who 0 qkrfmbw 4 asaupke

Cauchy, a famous mathematician, only gave a conjecture (the literature without his relevant proof still exists): the complex variable function f(z) is analyzed everywhere in the simply connected region B, then the integral of the function f(z) along any closed curve C in B is equal to 0. And about "How did Cauchy guess?" Looking back on this period of history, it is reasonable for Cauchy to make this assertion of Cauchy integral theorem. At that time, Cauchy's research on complex variable function was quite in-depth. He knew when f(z) was an analytic function, that is to say, when the line integral of f(Z) had nothing to do with the path (related C-R function papers were given in the form of attachments). Now, in order to vividly explain what I want to say, the attached drawings are as follows:

From the diagram above, Cauchy judged (at that time, he compared with functions) that if the blue curve is C 1 and the orange curve is C2, then the closed curve from A to B and then from B to A can be defined as C. Obviously, C=C 1+C2(-), where C 1 stands for A to B and C2(-). Then the following equation holds:

At this point, I think Mr. Cauchy almost has the idea of this conjecture. Mr. Cauchy gave some corresponding conditions in his conjecture because of the phenomena reflected in the integral calculation of specific complex variable functions, and Mr. Cauchy summarized them. Based on the above conditions, all conjectures of Cauchy theorem about Cauchy integral are collected, and the "dragon", which we now know as Cauchy-Gulsat law, is summoned.

This answer refers to some literatures and puts forward my own views. Personally, I feel that it should be a little close to the specific situation of Mr. Cauchy's conjecture, hoping to help you understand the historical evolution of Cauchy-Gulsat theorem and how great mathematicians think and put forward conjecture.