An Introduction to Higher Mathematics 2 Volume Set

The self-taught mathematician Hua Loo-Keng (1910–1985) has been credited with inspiring generations of mathematicians, while his papers on number theory are regarded as some of the most significant contributions made to the subject during the first half of the twentieth century. An Introduction to Higher Mathematics is based on the lectures given by Hua at the University of Science and Technology of China from 1958. The course reflects Hua's instinctive technique, using the simplest tools to tackle even the most difficult problems, and contains both pure and applied mathematics, emphasising the interdependent relationships between different branches of the discipline. With hundreds of diagrams, examples and exercises, this is a wide-ranging reference text for university mathematics and a testament to the teaching of one of the most eminent mathematicians of his generation.

Volume I: 1. Real and complex numbers; 2. Vector algebra; 3. Functions and graphs; 4. Limits; 5. The differential calculus; 6. Applications of the derivative; 7. Taylor expansions; 8. Approximate solutions to equations; 9. Indefinite integrals; 10. Definite integrals; 11. Applications of integral calculus; 12. Functions of several variables; 13. Sequences, series and integrals with variables; 14. Differential properties of curves; 15. Multiple integral; 16. Curvilinear integral and surface integral; 17. Scalar field and vector field; 18. Differential properties of curved surfaces; 19. Fourier series; 20. System of ordinary differential equations. Volume II: 1. Geometry of the complex plane; 2. Non-Euclidean geometry; 3. Definitions and examples of analytic and harmonic functions; 4. Harmonic functions; 5. Some basic concepts in point set theory and topology; 6. Analytic functions; 7. Residues and their application to definite integral; 8. Maximum modulus principle and the family of functions; 9. Entire function and meromorphic function; 10. Conformal transformation; 11. Summation; 12. Harmonic functions under various boundary conditions; 13. Weierstrass' elliptic function theory; 14. Jacobi's elliptic functions; 15. Systems of linear equations and determinants (review outline); 16. Equivalence of matrices; 17. Functions, sequences and series of square matrices; 18. Difference equations with constant coefficients and ordinary differential equations; 19. Asymptotic property of solutions; 20. Quadratic form; 21. Orthogonal groups and pair of quadratic forms; 22. Volumes; 23. Non-negative square matrices. Volume III: 1. The geometry of the complex plane; 2. Non-Euclidean geometry; 3. Definitions and examples of analytic functions and harmonic functions; 4. Harmonic functions; 5. Point set theory and preparations for topology; 6. Analytic functions; 7. The residue and its application to evaluation of definite integrals; 8. Maximum modulus theorem and families of functions; 9. Integral functions and metamorphic functions; 10. Conformal transformations; 11. Summability methods; 12. Harmonic functions satisfying various types of boundary conditions; 13. Weierstrass elliptic function theory; 14. Jacobian elliptic function theory. Volume IV: 1. Linear systems and determinants (review); 2. Equivalence of matrices; 3. Functions, sequences and series of square matrices; 4. Difference and differential equations with constant coefficients; 5. Asymptotic properties of solutions; 6. Quadratic forms; 7. Orthogonal groups corresponding to quadratic forms; 8. Volumes; 9. Non-negative square matrices.

Hua Loo-Keng (1910-1985) is remembered as one of the leading mathematicians of his generation. These volumes, containing both pure and applied mathematics, are based on his lectures at the University of Science and Technology of China. With hundreds of diagrams, examples and exercises, this is a wide-ranging reference text for university mathematics.

Hua Loo-Keng (1910–1985), a self-taught mathematician, is remembered as one of the leading scholars of his time. Hua spent most of his working life in China and suffered at first hand the turbulence of twentieth-century Chinese politics – but he also travelled extensively. This included time spent at Cambridge in the 1930s, when he made notable contributions to number theory, and post-war visits to Russia and America. Hua was appointed a professor of mathematics at the University of Illinois in 1949, but he chose instead to return to China to train the next generation of mathematicians and became the first director of the Mathematical Institute of the Academia Sinica. He was later appointed vice-president of Academia Sinica and a science advisor to his government. He continued to study and lecture abroad until his death in Tokyo in 1985. Hua received honorary degrees from the University of Nancy (1980), the Chinese University of Hong Kong (1983) and the University of Illinois (1984). His work has been translated into many languages, and Hua was elected a foreign associate of the National Academy of Sciences in 1982 and a member of the Deutsche Akademie der Naturforscher Leopoldina (1983), the Academy of the Third World (1983) and the Bavarian Academy of Sciences (1985).
Peter Shiu, now retired, was Reader in Pure Mathematics at Loughborough University. He has published widely on his own research interests and was the translator of Hua Loo-Keng's Introduction to Number Theory (1982) and Hua Loo-Keng by Wang Yuan (1998).