Asymptotics and Special Functions
This course provides an introduction to the theory and applications of asymptotic methods and special functions, which play a central role in modern analysis, applied mathematics, and mathematical physics. The course focuses on techniques for approximating functions, integrals, and solutions of equations in limiting regimes, as well as the study of classical special functions that arise in differential equations and mathematical modeling.
Topics in asymptotic analysis include asymptotic expansions, order relations, Laplace’s method, the method of steepest descent, and saddle-point techniques. Applications to integrals, series, and differential equations will be emphasized.
The course also introduces important families of special functions such as the Gamma and Beta functions, Riemann zeta function, Bessel functions, Legendre polynomials, Hermite polynomials, and hypergeometric functions. Their properties, integral representations, recurrence relations, and roles as solutions to classical differential equations will be explored.
Connections between asymptotic methods and special functions will be highlighted, including asymptotic expansions of special functions and their behavior in various limits.
By the end of this course, students should develop fundamental knowledge and skills involving basic concepts of the topics covered in this course. Overall, this course will serve as an essential ingredient for further graduate level courses in analysis and number theory, differential equations and even physics.
Topics in asymptotic analysis include asymptotic expansions, order relations, Laplace’s method, the method of steepest descent, and saddle-point techniques. Applications to integrals, series, and differential equations will be emphasized.
The course also introduces important families of special functions such as the Gamma and Beta functions, Riemann zeta function, Bessel functions, Legendre polynomials, Hermite polynomials, and hypergeometric functions. Their properties, integral representations, recurrence relations, and roles as solutions to classical differential equations will be explored.
Connections between asymptotic methods and special functions will be highlighted, including asymptotic expansions of special functions and their behavior in various limits.
By the end of this course, students should develop fundamental knowledge and skills involving basic concepts of the topics covered in this course. Overall, this course will serve as an essential ingredient for further graduate level courses in analysis and number theory, differential equations and even physics.
Lecturer
Date
15th September ~ 10th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 15:20 - 16:55 | Shuimo | - | - | - |
| Thursday | 13:30 - 15:05 | Shuimo | - | - | - |
Prerequisite
Topics covered: Euler's gamma and beta functions, hypergeometric functions, confluent hypergeometric functions, Bessel functions, Picard's theorems, Weierstrass factorization theorem, entire functions of finite order, summation formulae, asymptotics series, linear differential equations, Riemann zeta function, Benoulli numbers and polynomials, Bessel, Legendre and Whittaker functions.
Syllabus
Course Outline:
• Chapter 0. What are asymptotics and special functions? The big picture.
• Chapter 1. Introduction to asymptotic analysis.
• Chapter 2. Complex functions and integrals (Picard’s theorems, Weierstrass factorization theorem, entire functions of finite order).
• Chapter 3. Summation formulae and asymptotic series.
• Chapter 4. Introduction to special functions (Euler’s gamma and beta functions).
• Chapter 5. The Riemann zeta function.
• Chapter 6. Linear differential equations.
• Chapter 7. Hypergeometric functions.
• Chapter 8. Confluent hypergeometric functions (Bessel, Lengendre and Whittaker functions).
• Chapter 9. Orthogonal polynomials (Legendre, Chebyshev, Hermite and Laguerre polynomials)
• Chapter 0. What are asymptotics and special functions? The big picture.
• Chapter 1. Introduction to asymptotic analysis.
• Chapter 2. Complex functions and integrals (Picard’s theorems, Weierstrass factorization theorem, entire functions of finite order).
• Chapter 3. Summation formulae and asymptotic series.
• Chapter 4. Introduction to special functions (Euler’s gamma and beta functions).
• Chapter 5. The Riemann zeta function.
• Chapter 6. Linear differential equations.
• Chapter 7. Hypergeometric functions.
• Chapter 8. Confluent hypergeometric functions (Bessel, Lengendre and Whittaker functions).
• Chapter 9. Orthogonal polynomials (Legendre, Chebyshev, Hermite and Laguerre polynomials)
Reference
[1] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974; reprinted by AK Peters, 1997.
[2] E. D. Rainville, Special Functions, Macmillan, 1960.
[3] R. Wong, Asymptotic Approximations of Integrals, SIAM, 2001.
[4] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, 1999.
[5] C. Viola, Special Functions, Lecture Notes / Monograph.
[6] N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981.
[7] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978.
[8] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944.
[9] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927.
[10] A. Erdélyi et al., Higher Transcendental Functions, McGraw-Hill, 1953.
[11] NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/.
[12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1965.
[13] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
[14] H. M. Edwards, Riemann’s Zeta Function, Dover, 2001.
[15] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, 1986.
[2] E. D. Rainville, Special Functions, Macmillan, 1960.
[3] R. Wong, Asymptotic Approximations of Integrals, SIAM, 2001.
[4] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, 1999.
[5] C. Viola, Special Functions, Lecture Notes / Monograph.
[6] N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981.
[7] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978.
[8] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944.
[9] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927.
[10] A. Erdélyi et al., Higher Transcendental Functions, McGraw-Hill, 1953.
[11] NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/.
[12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1965.
[13] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
[14] H. M. Edwards, Riemann’s Zeta Function, Dover, 2001.
[15] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, 1986.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Cezar Lupu obtained his PhD degree from the University of Pittsburgh in 2018 with a thesis on special values of Riemann zeta and multiple zeta functions under the supervision of Piotr Hajlasz and William C. Troy. Between 2018-2021, he was a postdoctoral scholar at Texas Tech University under the mentorship of Razvan Gelca and Dermot McCarthy. In 2021, he moved to China as a postdoctoral fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) and Tsinghua University under the mentorship of Shing-Tung Yau until 2024. His main research interests are in the areas of number theory, analysis and special functions. Most of his recent research is centered around special values of L-functions and multiple zeta functions which play an important role at the interface of analysis, number theory, geometry and physics. He taught numerous courses at Pitt and TTU both undergraduate and graduate ranging from calculus and linear algebra to abstract algebra and real analysis. Also, he coached the best undergraduate students for the William Lowell Putnam Mathematical Competition. After moving to China, he taught courses at the Qiuzhen College, Tsinghua University. Together with other colleagues from Tsinghua University, he is organizing the Shadow Putnam Mathematical Competition at the Qiuzhen College. Moreover, starting 2023, he is the academic director of the International Mathematics Summer Camp (IMSC).