||Summary and further applications of the Residue Theorem; Short review of the midterm; Laurent series and the classification of singularities.
||The Residue Theorem; Computing residues; Examples; Summary; Application: evaluating convergent series.
||Application of the maximum modulus principle to count zeros; The fundamental theorem of algebra; Taylor series; Holomorphic functions are analytic; Zeros, poles and their multiplicities; The residue.
||Cauchy's integral formula (updated notes); Discussion and an example; Holomorphic functions are infinitely differentiable; The mean value theorem; The maximum modulus principle; The case of harmonic functions.
||Cauchy's theorem (Notes and a proof to be found here); Examples; Review of the midterm.
||Parametrized curves; Definition and examples; Integration of complex functions; Definition and examples; An upper bound; The case of functions with an antiderivative; Path independence.
||Trigonometric and hyperbolic functions; The multivalued logarithm; The principal branch of general powers; The multivalued square root; Outlook on Riemann surfaces; Dynamics of driven oscillating systems.
||The Cauchy-Riemann equations; Conjugate harmonic functions; Examples; The exponential and the logarithm.
||More on the complex exponential; Subsets: bounded, open connected; Complex functions; Examples; Continuity: Definition and examples; Differentiability; Holomorphic functions.
||Introduction; The set of complex numbers; The complex conjugate, inverses; The absolute value and the argument; The complex exponential; Trigonometric identities.