Li's achievements in mathematical research mainly include three aspects: sharp cone, piling up and prime number theory. Before analytic geometry and calculus were introduced to China, the theory of cone-tip technology was mainly written in three works, namely, Fiona Fang's Interpretation, Revealing the Secret of Arrow and Logarithmic Exploration, about 1845. The concept of "sharp cone" founded by Li is a geometric model to deal with algebraic problems. His description of "sharp cone curve" is essentially equivalent to giving equations such as straight line, parabola and cubic parabola.
He created the quadrature of a cone. It is equivalent to the formula of definite integral and the law of item-by-item integration of power function. He independently obtained the power series expansion of binomial square root by "element separation" and obtained the infinite series expression of π by combining with "quadrature of sharp cone". The expansions of various trigonometric functions and inverse trigonometric functions, and the expansions of logarithmic functions.
Using calculus to deal with mathematical problems has achieved creative results. The superposition theory is mainly found in the superposition ratio class, which was written in the period of 1859 ~ 1867. This is a work about high-order arithmetic progression. Li started with the study of China's traditional accumulation problem, and made some achievements equivalent to modern combinatorial mathematics. For example, "Triangular Stack Product Finding High Square Inexpensive Angular Table" and "Square Stack Inexpensive Table" are essentially the first famous Stirling numbers and Euler numbers in combinatorial mathematics. This is the famous "Li Identity" at home and abroad.
Since the 1930s, it has been widely concerned and appreciated by the international mathematics community. It can be considered that "stack-than-class" is the representative work of early combinatorial theory. The theory of prime numbers is mainly found in "Looking Up Numbers" published by 1872, which is the earliest work of prime number theory in China. When judging whether a natural number is a prime number, Li proved the famous Fermat prime number theorem and pointed out that its inverse theorem was not valid.
Li made outstanding contributions to the spread of classical mechanics in China. He translated The Course of Elementary Mechanics (second edition, 1833) written by the Englishman W. william whewell into Chinese (dictated by Ai), and it was published by Shanghai Mohai Library on 1859 with the title of "Re-learning", with a total of 20 volumes. This is the first Chinese translation that systematically introduces mechanics.
In the preface written by Li for the book Re-learning. Special emphasis is placed on the content of dynamics: "Push for a while and learn from it like shelling the enemy; Pushing for a long time is like five stars circling the sun, the moon circling the ground, and then learning. " "There are three rates of dynamic relearning: strength, quality and speed. If the strength is the same, the speed of people with small mass is high, and the speed of people with too large mass is low; If the quality is the same, the strength is small and the mystery is small, and the strength is high and the speed is fast. " "Dynamic relearning is pushing speed. Everything cannot move automatically, but it has the power to move. If it does not apply force after moving, it will move at a uniform speed. If you apply a constant force after moving, you will gradually accelerate. " "When everything rotates, it will ring the center of gravity and move. When two things are connected and intertwined, they will encircle the center of gravity and the moon and the earth will move together. "Li and William Williamson jointly translated the Outline of Astronomy written by the British astronomer Herschel (1792 ~ 187 1). The Chinese translation is called "Talking about Heaven" and published in 1859. Li made some omissions in his writing. This book not only systematically introduces modern astronomy to China for the first time. The theory of universal gravitation and celestial mechanics are also introduced. Chinese translation of some mechanical terms, such as perturbation and nutation. It was first seen in Chatting with Heaven. Since then, Li has been translating Newton's Mathematical Principles of Natural Philosophy. Although this book has not been completely translated, it has been lost after translation. However, since Li introduced Newtonian mechanics to China, the ideological system, viewpoints and methods of western modern science and some achievements in the history of modern science have brought new knowledge to China scholars, and at the same time aroused the enthusiasm of China scholars to learn natural science.
In "Real Gun and Live Gun Tactics", Li put forward a method of trajectory diagram according to the range formula of projectile on plane or inclined plane without considering air resistance. Although these achievements were lower than the ballistics level in Europe at that time, they reflected the process of natural science from introduction to digestion.