Learning objectives
<br /><br />To provide the basic instruments for the knowledge of successions and series of real or complex functions and the related types of convergence. In particular Fourier series.<br />Fornire strumenti per l'utilizzo delle funzioni di variabile complessa e dei loro integrali. In particolare serie di Taylor e di Laurent, teorema dei residui e definizione della trasformata di Fourier.<br />To provide the basic instruments for the knoledge of the complex variable functions and the related integrals. In particular the Taylor and the Laurent series, the residue Theorem and the definition of the Fourier transform.<br />
Prerequisites
<br />Calcolo I<br />Calcolo II<br />Calcolo III
Course unit content
<br />1. Successions of functions. Punctual convergence. Uniform convergence. Cauchy's criterion. Theorem of boundedness. Theorem of change of limits (n.p.). Theorem of continuity. Theorem of integrability (n.p.). Theorem of derivability.<br />2. Series of functions. Punctual, uniform and absolute convergence. Cauchy's criterion. Cauchy's N.C. Total convergence. Weierstrass' criterion. Theorems of boundedness, continuity, integradility, derivability.<br />3. Complex numbers. Cartesian, polar and exponential forms. Complex functions.<br />4. Holomorphic functions. Complex derivative. Cauchy-Riemann conditions. Confront with the real differentiability. De l' Hopital's theorem (n.p.).<br />5. Power series. Radius of convergence. Term by term derivability. Abel's criterion. Taylor's series. Expansion of elementary functions.<br />6.Fourier series. Punctual convergence. Uniform convergence. Quadratic mean convergence. Bessel's inequality. Parseval's identity. Fischer-Riesz theorem.<br />7. Countour integrals. Cauchy's theorem. Cauchy's integral representation formula. Mean value theorem. Maximum principle's theorem. Fundamental theorem of Algebra. Existence of a primitive.. Morera's theorem. Liouville's theorem.<br />8. Laurent's series. Isulated singularities : classification and characterization. Isulated singularity at infinity. Residue in a point and at infinite. The Cauchy's residues theorem.<br />9. Principle value of improper integrals. Great circle lemma. Jordan's lemma. Fourier 's transform.
Full programme
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Bibliography
1) Barozzi-Matarazzo, " Metodi Matematici per l'Ingegneria", ed. Zanichelli<br />2) Pagani-Salsa, " Analisi matematica II", ed. Masson<br />3) Spiegel " Analisi Complessa", collana Schaum's<br />4) Appunti del docente reperibili al centro fotocopie del Dip. Fisica
Teaching methods
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Assessment methods and criteria
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Other information
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