Syllabus
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1. Introduction to the complex analysis: Holomorfic function, Cauchy-Riemann equations, line integral in the complex domain, primitive function. Cauchy theorem, Cauchy formula, Liouville theorem. Taylor series, function holomorfic between circular contours, isolated singularities, Laurent series. Residue and Residue theorem. *
2. Fourier transform of functions Definition and basic properties. Schwartz space, L1 and L2 theory, inversion theorems, convolution, application to ODE and PDE. *
3. Distributions space D(Ω), topology, continuous linear functionals on D(Ω), order of distributions, convergence on D′(Ω), support of distributions, characterization of distributions of order 0 and non-negative distributions, derivative of distributions and its properties, approximation of δ-distributions by functions, Fourier series, Poisson summation formula, composition of distributions with diffeomorfisms, distributions with compact and point support, homogeneous distributions and their normalization. *
4. Tempered distributions, integral transform of distributions space of tempered distributions, convergence on S(R^N) and on S′(R^N), Fourier transform of tempered distributions, basic properties, tensor product of distributions and tempered distributions, convolution of distributions and tempered distributions, their Fourier transform, non-integer derivative, Fourier transform of selected distributions, Paley–Wiener theorem and its consequence, Fourier transform of radially symmetric funtions and distributions, surface measure.
Annotation
Basic mathematics course for 2nd year students of physics. Prerequisities: Mathematics for physicists I, NOFY161.