Modulets indhold, forløb og pædagogik

Purpose:
The purpose of this course is to provide the students with knowledge of, and to support the students in their understanding of, mathematical theories and methods of general applicability within the analysis of linear systems on an application level. In addition the course supports the students in their understanding of complex function theory and vector analysis.

Læringsmål

Viden

• demonstrate an understand of concepts, theories and methods used within the area of complex function theory, including:
  • analytical functions and their derivatives
  • Cauchy-Riemann equations
  • curve integrals
  • Cauchy's integral theorem and integral formula
  • graphical representations of standard complex mappings; Möbius (and its special cases), trigonometric, polynomial, logarithm, and exponential
• demonstrate an understanding of concepts, theories and methods used within the area of series theory and Fourier transformation, including:
  • Sequences, Series, Convergence Tests
  • Power Series - the coefficients and the center
  • Radius of Convergence R - Cauchy-Hadamards formula
  • Taylor and Maclaurin power series
  • Fourier Series for periodic funktions
  • Fourier Series for even and odd functions
  • Fourier Cosine and Fourier Sine Series
  • Fourier Integrals
  • The Fourier Transform
  • amplitude and phase specters by the Fourier Transform
• demonstrate an understanding of concepts, theories and methods used within the area of vector analysis, including:
  • skalarfelter og vektorfelter
  • rumlige integraler, herunder kurveintegraler, fladeintegraler og volumenintegraler i forskellige varianter
  • begreberne flux og cirkulation
  • rumlig differentialer, herunder gradient, divergens og rotation
  • parametriske beskrivelser af kurver og flader
  • Greens sætning. Stokeses sætning, Gausses sætning og Holmholtzes sætning
  • begreberne konservative felter og solenoidale felter
  • begrebet potentialefunktion

Færdigheder

• apply the presented concepts, theories and methods used within the area of complex function theory to:
  • determine function properties; continuity and analyticity
  • apply Cauchy-Riemann equation to funktions to determine of a function is analytical
  • relation between exponential and trigonometric and hyperbolic functions
  • Möbius transform and its special cases, including dilation, translation, rotation and inversion
  • design of Möbius transform based on mapping points
  • curve integrals, closed curve integrals, and parameterization of these
  • path-independent curve integrals
  • finding critical points for functions
  • apply Cauchy's integral theorem and formula to analytical functions
• apply the presented concepts, theories and methods used within the areas of series theory and Fourier transformation to:
  • Series analysis with special focus at convergence test (e.g. by Comparison Test, by Ratio Test or by Root Test)
  • Specification and analysis of Power Series with special focus at convergence and calculation of the Radius of Convergence R by Cauchy-Hadamards formula
  • Power Series development by Taylor and Maclaurin approximation
  • development of Fourier Series for periodic funktions
  • delvelopemtn of Fourier Series for even and odd functions - and for arbitrary periods (2L)
  • development of Fourier Integrals
  • development of the Fourier Transformation for real and complex functions
  • calculate amplitude specters and phase specters for Fourier Series and Fourier Transforms
• apply the presented concepts, theories and methods used within the area of vector analysis to:
  • fremstille parametriske repræsentationer af kurver og flader ud fra verbale, formelle eller grafiske beskrivelser (- en tegning!)
  • skitsere givne kurver og flader
  • evaluere kurveintegraler, dobbeltintegraler, fladeintegraler og volumenintegraler
  • bestemme divergens, gradient og rotation for givne skalar- og vektorfelter
  • evaluere rumlige integraler under anvendelse og Gausses sætning og stokeses sætning
  • bestemme en potentialfunktion for et givne konservativt felt samt kontrollere løsningen
  • evaluere vejuafhængige kurveintegraler ved at finde stamfunktion

Kompetencer

• based on given prerequisities, to reason for design choices and to enter into discussions regarding linear systems using the terminology of complex function theorym series theory and Fourier transformation, and vektor analysis
• use relevant concepts, theories and methods within complex function theory to:
  • Determine the correct method of integration for given functions
  • Determine in which domain a given function in analytical
  • Recognize the specific transforms Möbius (and its special cases), trigonometric, polynomial, logarithm, and exponential
  • Present solutions to problems in a clear and concise fashion
• use relevant concepts, theories and methods within series theory and Fourier transformation to:
  • perform analysis of Series and the related Convergence
  • make appropriate choices for Taylor-/Maclaurin Series approximations e.g. on the center and the amount of coefficients
  • make appropriate choices for Fourier Series approximation for periodic functions e.g. concerning the fundamental period length, symmetry and the amount of coefficients (covering the spectral bandwidth)
  • determine the use Fourier Transformation - especially for spectral analysis i.e. for calculation of the amplitude spectra, the phase spectra and the power spectra
• use relevant concepts, theories and methods within vector analysis to:
  • vurdere en given opgave i vektoranalyse og udvælge den mest hensigtmæssige løsningsform
  • fremstille løsningen således, at tankegangen klart fremgår på en saglig måde.

Undervisningsform

Forelæsninger, opgaveregning, workshops, selvstudie.

Eksamen

Prøver