Mathematics: Complex Analysis 1

<p>The course provides a solid introduction to the theory of analytic functions and provides concepts and methods from complex analysis that are important for further studies in mathematics and other areas of science.</p><p>The course treats basic theory of analytic functions:</p> <ul> <li>Elementary properties of analytic functions in one variable. Complex differentiability and Cauchy-Riemann equations. Calculation rules. Elementary examples of analytic functions: power series expansions, exponential functions, branches of logarithms, and functions defined by these calculation rules.</li> <li>Contour integrals in the complex plane. Cauchy’s integral theorem and integral formula. Existence of a primitive function and local power series expansion of analytic functions. Cauchy estimates, Liouville’s theorem, and the fundamental theorem of algebra.</li> <li>Theory of meromorphic functions: singular points, Laurent series expansion and the residue theorem. Residue calculus.</li> <li>Further elements of the theory of holomorphic functions such as argument principle, Rouché’s theorem, and open mapping property.</li> <li>Harmonic functions. Regularity, existence of harmonic conjugate, mean value property, maximum principle, Poisson integrals.</li> </ul>