Perturbation Theory and the WKB method

Course information

Code: MAT1312F
Instructor: Marco Gualtieri
Class schedule: R2-5 BA6183 [starting Sept. 18]
Evaluation: Exegesis (or Eisegesis) of one of the articles listed below, or another approved by the instructor. Some additional exercises. Attendance.

Recently there has been a surge of interest in topics such as the Stokes phenomenon, the WKB approximation, the theory of Borel summability of divergent series, and the study of quadratic differentials on a Riemann surface, because of the ways in which these topics illuminate the study of quantum field theories and the meaning of the Feynman diagram series in perturbation theory. The basic aim of this course is to explain perturbation theory and to explore how to go “beyond perturbation theory”.

While we will discuss issues at the forefront of research, we will not assume any background in the area: the only prerequisite is an understanding of basic complex analysis, differential equations and differential geometry, such as is usually covered at the undergraduate level.



The main book we will use is the beautiful book of Kawai and Takei entitled “Algebraic Analysis of Singular Perturbation Theory”, available from the AMS here. Other useful references include:

  • Olver’s book “Asympotics and Special Functions”
  • Wasow’s book “Asymptotic expansions for ordinary differential equations”
  • Mark Holmes: “Introduction to Perturbation Methods”
  • Bender and Orszag text: “Advanced Mathematical Methods for Scientists and Engineers”