#### Objective

- knowledge: To train the students to develop a coherent mathematical theory.
- know-how: To train the students to be been useful of this theory in some fields of application practical: calculations of some real integrals without recourse to the primitives, resolution of the linear differential equations of the second order around a point of singularity, deformations of the fields, initiation to the analytical theory of the numbers: theorem of the prime number, etc…

#### Content

1. Algebra of the formal series C [[X]].
2. Complement on the power series: Problem of convergence on the border of the disc of convergence, differentiability of the sum of a power series, to redefine the elementary functions with the means of the power series, surfaces of Riemann and some multiform functions, in particular various determinations of the function logarithm, concept of angle.
3. Complement of integration: Exact differential forms, closed, theorem of Cauchy - Goursat, theorem of Morera. Concept of homotopy, primitive of a closed differential form according to a path, integration of a closed differential form on two homotopic paths, simply connected domains in C.
4. Cauchy integral formula: Index of a closed loop with respect to a point, general form of the Cauchy integral formula, representation of a holomorphic function by a Taylor series in a disc, relation between analycity and holomorphy, inequalities of Cauchy, theorem of Liouville, concept of singularity, representation by a series of Laurent of a holomorphic function around a singular point, theorem of Weierestass around an isolated essential singular point, general information on the residue theorem in a domain containing poles, essential isolated singular points..
5. Principle of the analytical prolongation: Numbers of zeros and poles lying inside a closed contour, theorem of unicity of the analytical prolongation, principle of symmetry of Schwartz.
6. Harmonic functions: functions that have the property of the average, principle of maximum, harmonic functions and their prolongation, Kernel of Poisson, problem of Dirichlet on the disc. Characterization of the harmonic functions.
7. Conformal transformations: Necessary and sufficient condition so that a holomorphic transformation is conformal, homographic functions, some examples. Statements of the theorem of Riemann characterizing the simply connected domain.
8. Topology of the compact convergence of H(Ω). Theorem of the open application, uniform compact convegence, associated topology, relation between the zeros of functions which converges uniformly and those of its limit for the holomorphic or meromorphic functions, infinite products of holomorphic functions and their convergence.
9. Some applications in applied mathematics or physics: One or two examples drawn from the heading know-how, noted before, according to remaining time available.