Educational guide 2023_24 School of Engineering

Contents

Topic	Sub-topic
The complex plane	The field of the complex numbers. Modulus and argument. DeMoivre's Formula. Exponential and logarithm of a complex number. Estereographic projection. Basic topology of the complex plane.
Holomorphic function	Definition of a holomorphic map. Derivative rules. Cauchy-Riemann Equations. Examples of holomorphic maps. Power series. Weierstrass M-test. Radius of convergence.
Curves in the complex plane	The parametrization of a curve. Different types of curves. Recitifiable curves. Piecewise C1 curve. Orientation. Length. Index of a closed curve. Cyclea.
Complex Integration and Cauchy's Theorems	Line integrals. Primitive function. Cauchy's Theorem for a triangle. Cauchy's Theorem for a convex domain. Cauchy's Theorem for a simply connected domain. Cauchy Theorem for a cycle.
Consequences of Cauchy's Theorem	Taylor's Theorem. Power series expancion. Cauchy's estimate. Liouville's Theorem. Fundamental Theorem of Algebra. Morera's Theorem. Maximum modulus principle. Open mapping Theorem.
Singularities of an holomorphic mapping and residue Theorem	Classification of singularities: removable, pole and esential. Laurent serie expansions. Residue of a singularity. Calculating residues. Residue Theorem. Applications.
Further topics	Fourier series expansions. Riemann mapping Theorem. Riemann zeta function.