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# VPM

Course: Selected Lessons in Mathematics

Department/Abbreviation: OPT/VPM

Year: 2020

Guarantee: 'doc. Mgr. Ladislav Mišta, Ph.D.'

Annotation: Algebra of complex numbers Progressions and series

Course review:
Algebra of complex numbers Progressions and series Function of the complex variable Limit and continuity of the complex function Complex function of the real variable Curves in the complex plane Differentiation of the complex function Holomorphic functions Progressions and series of complex functions Power series Elementary functions of the complex variable Contour integral of the complex function Cauchy theorem Cauchy formula and integral of Cauchy type Primitive functions Index of the point with respect to the contour Taylor series of the holomorphic function Total function Laurent series of the function holomorphic in the ring Isolated singular points of the holomorphic function and their classification Residuum of the function in the point Residuum theorem Use of the residuum theorem for the calculation of the integrals Jordan lemma

Integral transforms

Introduction, motivations: laws of thermomechanics, derivation of system of equations of nonlinear theory of bounded thermoelasticity, linearization, simplification, elasticity, heat conduction, idea of transformation of partial differential equation into ordinary differential equations by Fourier transform Formalization: abstract Hilber spaces, Fourier series, properties, examples, use Application: spaces of smooth integrable functions, distributions, functions with finite energy, Sobolev spaces, dual spaces, duality, interpretations in mechanics Fourier transform: definition, properties, examples, use of Fourier transform, definition of Sobolev spaces by means of Fourier transform, Fourier-Poisson integral, Green function, practical applications, heat conduction, examples Laplace transform: definition, properties, applications, examples