Lecture 1: Introduction to complex numbers, functions of complex variables, Cauchy-Riemann conditions. Chapter 1&2 of the book by Carrier et al. or Chap. 6 of the book by Arfken et al.
Lecture 2: Branch points and branch cuts; analytic functions. Chapters 1&2 of Carrier et al. or Chap. 6 of the book by Arfken et al.
Lecture 3: Integration in the complex plane; Cauchy Theorem; Cauchy Integral Formula and higher-order derivatives. Chapter 2 of Carrier et al. or Chap. 6 of the book by Arfken et al.
Lecture 4: Cauchy Integral Formula for Laplace equation with Dirichlet and Neumann boundary conditions. Chapter 2 of Carrier et al.
Lecture 5: Taylor and Laurent expansions. Example 6.6.1 Arfken et al. book. Material is also covered in Chapter 2 of Carrier et al. book. Isolated singularities, poles, residues and essential singularities. Material is covered in Carrier et al. and Arfken et al. books.
Lecture 6: Contour integrals. Material is covered in Carrier et al. and Arfken et al. books.
Lecture 7: Asymptotic expansions and their properties. Use of integration by parts. Material is covered in Carrier et al. and Bender and Orszag (Chaps. 3 & 6).
Lecture 8: Laplace method, Watson Lemma. Material is covered in Carrier et al. and Bender and Orszag (Chap. 6).
Lecture 9: Examples of Laplace method and introduction to Method of Stationary Phase. Material is covered in Carrier et al. and Bender and Orszag (Chap. 6).
Lecture 10: Method of Stationary Phase and Introduction to Steepest Descent. Material is covered in Carrier et al. and Bender and Orszag.
Lecture 11: Method of Steepest Descent. Material is covered in Carrier et al. and Bender and Orszag.
Lecture 12: Classification singular points ODEs. Material is covered in Bender and Orszag.
Lecture 13: Asymptotic analysis for irregular singular points. Material is covered in Bender and Orszag.
Lecture 14: Sturm-Liouville. Material is covered in Arfken's book.
Lecture 15: Green's function methods. Material is covered in Arfken's book.
Lecture 16: Laplace transform. Material is covered in Arfken's book.
Lecture 17: Applications of Laplace transform to ODEs; Boundary layers and introduction to WKB method. Material on WKB will follow Bender and Orszag presentation.
Lecture 18: WKB method. Chap. 10 Bender and Orszag.
Lecture 19: Turning points and connection formulae; Quantization of energy levels. Chap. 10 Bender and Orszag.
Lecture 20: Multiple-scale methods; nonlinear oscillators and stability of Mathieu equation. Chap. 11 Bender and Orszag.