Computer engineering (Academic Year 2019/2020) - Computer Engineering

Advanced Mathematical Methods

Lesson n. 1: Course overview Contents Complex function theory Working with functions and integrals A transform to the rescue Using complex numbers		Simon Salamon
Lesson n. 2: Using complex number Contents Representing complex numbers Exponents and conjugates Applications to roots and powers Functions of a complex variable		Simon Salamon
Lesson n. 3: Holomorphic functions Contents Mappings from the complex plane to itself Limits of a complex variable Complex derivatives Polynomial functions		Simon Salamon
Lesson n. 4: The Cauchy Riemann equations Contents Partial derivatives Recognizing holomorphic functions Vector calculus Harmonic functions		Simon Salamon
Lesson n. 5: Power series Contents Polynomials and series Absolute convergence Radius of convergence Analytic functions		Simon Salamon
Lesson n. 6: Contour integration Contents Paths and curves The velocity integral Integrals along curves Length estimates		Simon Salamon
Lesson n. 7: Cauchy's theorem Contents Fundamental theorem of calculus Regions in the plane Existence of primitives Integrating a holomorphic function		Simon Salamon
Lesson n. 8: Cauchy's integral formula Contents Existence of primitives A singular integrand Integrating around circles Deforming a contour		Simon Salamon
Lesson n. 9: Laurent series Contents Taylor series Derivatives of holomorphic functions Series of reciprocals Convergence in an annulus		Simon Salamon
Lesson n. 10: Residues and boundaries Contents Laurent coefficients Simple closed contours Cauchy’s Theorem for boundaries Computing integrals using residues		Simon Salamon
Lesson n. 11: Singularities and integrals Contents Isolated singularities Poles and zeros Computing and using residues Trigonometric integrals		Simon Salamon
Lesson n. 12: Polynomials and definite integrals Contents Fundamental theorem of algebra Rational functions More integrals involving roots of unity More trigonometric integrals		Simon Salamon
Lesson n. 13: Further integration tecnique Contents Indented contours Semicircular estimates Logarithmic integrals Summation of series		Simon Salamon
Lesson n. 14: Laplace transforms Contents Basic properties Further examples Transforms of derivatives Solving an initial value problem		Simon Salamon
Lesson n. 15: Transforms calculus Contents Limiting values and integrals New transforms from old Some special integrals Applications to finding inverse transforms		Simon Salamon
Lesson n. 16: The inverse Laplace transforms Contents Inverting rational functions Known examples of inverse transforms Contour integral interpretation Applying the inversion theorem		Simon Salamon
Lesson n. 17: The theory of distributions Contents The set of test functions Linear functionals Distributions as limits Derivatives of distributions		Simon Salamon
Lesson n. 18: Working with distributions Contents Operations on distributions Differentiation and limits Principal value integrals Infinite sums and series		Simon Salamon
Lesson n. 19: Convolution of function Contents Motivating examples Convolution as a product Spaces of integrable functions Convolution with a distribution		Simon Salamon
Lesson n. 20: The Fourier transform Contents First examples Spectral analysis Derivatives and products Transform of a convolution		Simon Salamon
Lesson n. 21: Fourier inversion Contents The inversion theorem Proof of inversion Using a convolution The Schwartz space		Simon Salamon
Lesson n. 22: Fourier transforms of distributions Contents Convergence in Schwartz space Tempered distributions The generalized Fourier transform Generalized inversion		Simon Salamon
Lesson n. 23: Back to Laplace transforms Contents Laplace versus Fourier Laplace convolution An application to beam bending Laplace inversion		Simon Salamon
Lesson n. 24: Derivatives, series and integrals Contents Another differential equation Solution by series Laurent coefficients Laplace transform of an integral		Simon Salamon
Lesson n. 25: A final application Contents A partial differential equation The heat kernel Final example Acknowledgements		Simon Salamon

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Computer engineering (Academic Year 2019/2020) - Computer Engineering

Advanced Mathematical Methods

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Computer engineering (Academic Year 2019/2020) - Computer Engineering

Advanced Mathematical Methods

Slides