Università telematica internazionale UNINETTUNO

MOOC Massive Open Online Courses (Academic Year 2018/2019)

Complex Analysis


Leçon vidéo

Leçon n. 1: Course overview
   Contents

   Complex function theory

   Working with functions and integrals

   A transform to the rescue

   Using complex numbers
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Leçon n. 2: Using complex number
   Contents

   Representing complex numbers

   Exponents and conjugates

   Applications to roots and powers

   Functions of a complex variable
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Leçon n. 3: Holomorphic functions
   Contents

   Mappings from the complex plane to itself

   Limits of a complex variable

   Complex derivatives

   Polynomial functions
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Leçon n. 4: The Cauchy Riemann equations
   Contents

   Partial derivatives

   Recognizing holomorphic functions

   Vector calculus

   Harmonic functions
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Leçon n. 5: Power series
   Contents

   Polynomials and series

   Absolute convergence

   Radius of convergence

   Analytic functions
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Leçon n. 6: Contour integration
   Contents

   Paths and curves

   The velocity integral

   Integrals along curves

   Length estimates
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Leçon n. 7: Cauchy's theorem
   Contents

   Fundamental theorem of calculus

   Regions in the plane

   Existence of primitives

   Integrating a holomorphic function
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Leçon n. 8: Cauchy's integral formula
   Contents

   Existence of primitives

   A singular integrand

   Integrating around circles

   Deforming a contour
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Leçon n. 9: Laurent series
   Contents

   Taylor series

   Derivatives of holomorphic functions

   Series of reciprocals

   Convergence in an annulus
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Leçon n. 10: Residues and boundaries
   Contents

   Laurent coefficients

   Simple closed contours

   Cauchy’s Theorem for boundaries

   Computing integrals using residues
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Leçon n. 11: Singularities and integrals
   Contents

   Isolated singularities

   Poles and zeros

   Computing and using residues

   Trigonometric integrals
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Leçon n. 12: Polynomials and definite integrals
   Contents

   Fundamental theorem of algebra

   Rational functions

   More integrals involving roots of unity

   More trigonometric integrals
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Leçon n. 13: Further integration tecnique
   Contents

   Indented contours

   Semicircular estimates

   Logarithmic integrals

   Summation of series
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Leçon n. 14: Laplace transforms
   Contents

   Basic properties

   Further examples

   Transforms of derivatives

   Solving an initial value problem
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Leçon n. 15: Transforms calculus
   Contents

   Limiting values and integrals

   New transforms from old

   Some special integrals

   Applications to finding inverse transforms
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Leçon n. 16: The inverse Laplace transforms
   Contents

   Inverting rational functions

   Known examples of inverse transforms

   Contour integral interpretation

   Applying the inversion theorem
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Leçon n. 17: The theory of distributions
   Contents

   The set of test functions

   Linear functionals

   Distributions as limits

   Derivatives of distributions
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Leçon n. 18: Working with distributions
   Contents

   Operations on distributions

   Differentiation and limits

   Principal value integrals

   Infinite sums and series
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Leçon n. 19: Convolution of function
   Contents

   Motivating examples

   Convolution as a product

   Spaces of integrable functions

   Convolution with a distribution
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Leçon n. 20: The Fourier transform
   Contents

   First examples

   Spectral analysis

   Derivatives and products

   Transform of a convolution
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Leçon n. 21: Fourier inversion
   Contents

   The inversion theorem

   Proof of inversion

   Using a convolution

   The Schwartz space
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Leçon n. 22: Fourier transforms of distributions
   Contents

   Convergence in Schwartz space

   Tempered distributions

   The generalized Fourier transform

   Generalized inversion
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Leçon n. 23: Back to Laplace transforms
   Contents

   Laplace versus Fourier

   Laplace convolution

   An application to beam bending

   Laplace inversion
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Leçon n. 24: Derivatives, series and integrals
   Contents

   Another differential equation

   Solution by series

   Laurent coefficients

   Laplace transform of an integral
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Leçon n. 25: A final application
   Contents

   A partial differential equation

   The heat kernel

   Final example

   Acknowledgements
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00186 Roma - ITALIA
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Certified mail

info@pec.uninettunouniversity.net

Secrétariat des Etudiants

tel: +39 06 692076.70
tel: +39 06 692076.71
e-mail: info@uninettunouniversity.net

Vidéoconférence

Library 1st floor: 90.147.90.157
Meeting Room 5th floor: 90.147.90.158

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