Complex numbers, elementary operations: addition, multiplication, their properties. The conjugate, the absolute value and their behaviour with respect to addition and multiplication. Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses.
Open, closed, connected sets. Limits of sequences and functions, their behaviour with respect to addition, multiplication, division. Cauchy's criterion for convergence. Continuous functions. Continuity of sums, products, ratios, compositions. Definitions of continiuty using open and closed sets. Connectedness, its preservation under continuous maps. Uniform convergence and continuity of uniform limits of continuous functions. Branches of multi-valued functions. Examples of branches of the argument function and the logarithm.
Infinite series of complex numbers. Geometric series and its convergence properties. The comparison test. Absolute convergence. Infinite function series and their uniform convergence. Weierstrass test. Power series. Abel's Lemma. Radius of convergence.
Complex-differentiable and holomorphic functions. Differentiability of sums, products, ratios, composition and inverse functions. Real-differentiable functions. Cauchy-Riemann equations. Complex differentiability of polynomials, rational functions, exponential, logarithm and trigonometric functions.
Path integrals. Independence of parametrization. Length of a path and estimates for path integrals. Antiderivatives. Calculation of path integrals using antiderivatives.
Cauchy's theorem: Goursat's version for a triangle, generalization for polygonal regions and simple bounded regions. Cauchy's integral formula. Residue theorem. Calculation of residues for ratios of holomorphic functions.
Applications of Residue theorem: Trigonometric integrals, Improper integrals, Fourier transform type integrals.
Taylor series and Laurent series expansions. Differentiation of power series. Poles. Calculation of residues using Laurent series expansion. Order of zeroes and poles. Identity principle. Maximum modulus principle.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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