| Center |
| Faculty of Mathematics |
| Departament |
| Mathematical Analysis |
| Lecturers in charge |
| F6092 - JOSE M MAZON RUIZ |
| Met. Docent |
| Met. Avaluació |
| Final examination - - |
| Bibliografia |
| ASH, R.B. "Complex Variables". Academic Press 1971 APOSTORL, T.M. "Análisis matemático". Revert, 1976 BURCKEL, R.B. "An introduction to Classical Complex Anlaysis). Academic Press. 1979 CONWAY, J.B. "Functions of One Complex Variable".Springer. 1978 JAMESON, D.J.O. "primer Curso de Funciones Complejas". Compañia Editorial Continental. S.a. 1970 KNOPP, K. "teory of Functions".Dover. 1947 KOLMOGOROV, A.N. fomn, s.v. "Elementos de Teoría de Funciones y del Análisis Funcional" Mir. 1975. KRZYZ, J.G. "Problems in Complex Vriable Theory". American Elsevier Pub. Co., 1971 MARSDEN, J.E., HOFMAN, J.J. "Basic Complex Analysis" W.H.Freeman and Co. 1970 MARKUSEVICHE, A. "Teoría de las Funciones Analítica. (2 vols.)Ed. Mir. 91970 |
| Continguts |
| Goals The main goal of this course is to introduce the student into the theory of functions of a complex variable,showing its main properties and applications: Cauchy's Theorem, the Residues Theorem, as well as its application to evaluatingreal integrals and series. The last part of this matter is devoted to the study of harmonic functions and the Laplace transform. Theory programme Chapter 2. Power series.Power series. Radius of convergence. Cauchy-Hadamard's formula. Derivation of a power series. Products of power series. Dirichlet's criterium. Analytic functions. Principle of analytic. Chapter 3. Elementary functions. The exponential and trigonometric functions. Definition of argument and argument branch. Properties. Definition of logarithm and logarithm branch. Sufficient condition for the existence of logarithm branches. Chapter 4. Path integration. Integral along a path. Fundamental Theorem of Calculus. Functions defined by path integrals. Index of a closed path respect to a point. Chapter 5. Cauchy-Goursat Theorem and its consequences. Cauchy-Goursat Theorem. The avoidable singularity theorem. Cauchy's integral formula. Analiticity of holomorphic functions. Morera's theorem. Cauchy's inequalities. Weierstrass' theorem. Liouville's theorem. T |