Lecture 1: Introduction to complex numbers and series, radius of convergence, functions of complex variables. Chapter 1 of the book by Carrier et al. or Chap. 6 of the book by Arfken et al.
Lecture 2: Branch points and branch cuts; Differentiation in the complex plane and Cauchy-Riemann conditions. 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: Dirichlet and Neumann problems; harmonic functions and basic ideas of conformal mappings. Chapters 2&4 of Carrier et al.
Lecture 5: Taylor and Laurent expansions. Expansions for the Example 6.6.1 Arfken et al. book. Material is also covered in Chapter 2 of Carrier et al. book.
Lecture 6: Isolated singularities, poles, residues and essential singularities; analytic continuation. Example of the Euler Gamma function and review of its main properties. Material is covered in Carrier et al. and Arfken et al. books.
Lecture 7: Residue Theorem, Jordan lemma and first examples of contour integrals. Material is covered in Carrier et al. and Arfken et al. books.
Lecture 8: Contour integrals. Material is covered in Carrier et al. and Arfken et al. books.
Lecture 9: Asymptotic expansions and their properties. Use of integration by parts; Laplace method. Material is covered in Carrier et al. and Bender and Orszag (Chaps. 3 & 6).
Lecture 10: Watson's Lemma and other examples of Laplace method. Material is covered in Carrier et al. and Bender and Orszag (Chap. 6).
Lecture 11: Method of Stationary Phase. Material is covered in Carrier et al. and Bender and Orszag (Chap. 6).
Lecture 12: Method of Steepest Descent and Saddle point. Material is covered in Carrier et al. and Bender and Orszag (Chap. 6).
Lecture 13: Classification of singular points of homogeneous linear ODEs. Material is covered in Arfken and Bender and Orszag (Chap. 3).
Lecture 14: Frobenius series and asymptotic methods for irregular singular points. Method of dominant balance. Material is covered in Bender and Orszag (Chap. 3).
Lecture 15: Sturm-Liouville systems. Material is covered in Arfken's book.
Lecture 16: Non-homogeneous equations and Green's function methods for boundary and initial value problems. Material is covered in Arfken's book.
Lecture 17: Laplace transforms, applications to ODEs, inverse transform and its asymptotic expansion for large arguments. Material is covered in Arfken's book.
Lecture 18: WKB theory, matching and one turning point. Material is covered in Bender and Orszag book.
Lecture 19: Special functions and polynomials. Material is covered in Arfken's book.