The opening report is produced with the enhancement of the planning of modern scientific research activities and the need of programmed management of scientific research topics. The following is the model essay of the opening report of the Department of Mathematics that I compiled for you. Welcome to reading.

Title: A Comparative Study of Real Integral and Complex Integral

First, the source and significance of the topic.

Through the study of mathematical analysis and complex variable function, I know that a lot of knowledge in complex variable function theory is extended and expanded on the basis of mathematical analysis, and complex integration, to a great extent, broadens the variable range of real integral, that is, integrating in complex number field. Integral calculus is a complete mathematical theory based on the germination of ancient eastern and western calculus thoughts and the ideological and social background of Europe on the eve of the establishment of calculus, which was finally formed after many algebraic studies and explorations. The comparative study of real integral and complex integral is a subject worthy of my consideration and research.

Integral is an important content in function theory. Both real integral and complex integral are important tools for studying functions, which are widely used in geometry, physics and engineering technology. Complex integral is an important part of complex variable function theory, and its role in studying complex variable functions, especially analytic functions, far exceeds that of real integral in studying real variable functions. Whether in the study of complex variable function, differential, series or their various applications, the integration theory of complex variable function should be used. Complex integral is a generalization of real integral, which is calculated by complex integral. Therefore, comparing the properties and applications of complex integral and real integral is of great significance for deeply understanding the theory of complex variable function and using these theories to solve various practical problems in mathematics and other disciplines.

Second, the development and research background at home and abroad

Many mathematicians in China analyze and study integral calculus, and many university teachers also study complex integral and real integral. Wan Qiaoling, Department of Mathematics, Longdong University, has conducted a comprehensive study on "Calculating Real Integral with Complex Integral" and published related papers. Wang of Shaanxi Institute of Education also published the related paper "The Connection and Difference between Real Integral and Complex Integral". The study of integral calculus abroad is more extensive and far-reaching than in China. Real integral and complex integral are the concrete contents of integral calculus. Modern integral is different from the previous integral to some extent, but it is formed on the basis of the past and has been perfected by many algebras. Integral calculus originated from calculus (calculus originated from Newton and Leibniz), and the core concept of calculus is limit. The perfection of this theory is attributed to the work of Cauchy and Wilstrass in the19th century. /kloc-in the 0/7th century, the calculation of area and curve length by integral calculus began with Kepler, who published a new science of measuring the volume of wine barrels. Mathematicians such as Torricelli, Fermat, Pascal, etc. The shortcomings of the previous integral are repaired and perfected, which makes the integral closer to the modern integral. Integral is not only a tool to study functions, but also widely used in geometry, physics, engineering technology and other aspects.

III. Objectives and contents of the study

Through the comparative study of real integral and complex integral, we are familiar with and master the concepts and types of real integral and complex integral, classify and summarize them, find out the differences and connections between them, and understand the related applications of complex integral and real integral.

(1) The research background of the comparative study of real integral and complex integral, the present situation of this topic at home and abroad, and the significance of this study.

(2) Related concepts, properties and calculation methods of real integral and complex integral (definite integral and curve integral).

(3) Compare the definitions, properties, calculation methods and applications of real integral and complex integral; The relationship between real integral and complex integral (calculate real integral with complex integral and give an example to explain).

Fourthly, the research method of this topic.

This topic will make a comparative study of real integral and complex integral through analysis, comparison and synthesis, and finally illustrate that some real integral problems can be solved by using complex integral through examples.

V. Project progress plan:

The first stage: collect information, determine the scope of the topic, and contact the instructor (20XX autumn 1-7 weeks).

The second stage: selecting the topic, filling in the opening report and preparing for the opening (20XX autumn 8- 12 weeks).

The third stage: the instructor guides the investigation, collects data and prepares to write the first draft (20XX autumn 13 weeks-20XX spring 6 weeks).

The fourth stage: write the first draft and revise the paper under the guidance of the tutor (20XX spring 7- 14 weeks).

Stage 5: Submit the thesis, prepare the defense and summarize the thesis (20XX spring 15- 16 weeks).

Reference of intransitive verbs

[1] Zhong Yuquan. Complex variable function theory [M]. Third edition. Beijing: Higher Education Press, 2004.

[2] Department of Mathematics, East China Normal University. Mathematical analysis [M]. Third edition. Higher education press, 200 1

[3] Mathematics Department of Sichuan University. Advanced Mathematics (Volume IV) [M]. Beijing: Higher Education Press, 2002.

[4] Yan Ziqian, etc. Mathematical Analysis (I) [M]. Beijing: Higher Education Press, 2004

[5] Qinghua Sun, Zhao, The Solution of New Complex Variable Function [M]. Wuhan: Huazhong University of Science and Technology Press, 2002.

[6] Wang. The connection and difference between real integral and complex integral [N]. Journal of Shaanxi Institute of Education, 1995, 25:73-79

[7] Wan Qiaoling. Calculate real integral [N] with complex integral. Journal of Heze University, 20/kloc-0,32 (2):1673-2103.

[8], Wang, Prove real integral skillfully with complex function integral [J]. Mathematics Teaching and Research Examination Weekly, 2009,41.

Jin Yunjuan. Application of uniqueness theorem of analytic function in complex integral [N]. Journal of lishui university, March, 20091(5)

[10] Cui Dongling. Calculation method of complex integral [J]. Journal of Huainan Normal University, 2006,3: 6-9

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