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 understanding of concepts, theories and methods used within the area of complex function theory, including:analytical functions and their derivatives
• Cauchy-Riemann equations
• curve integrals
  •  
• 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:
  • sampling continuous time  functionsDiscrete Fourier Transformation and Fast Fourier Transformation
• 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 volumenintegralerflux og cirkulation.
  • Greens sætning, Stokeses sætning, Gausses sætning og Helmholtzes sætning

Færdigheder

• apply the presented concepts, theories and methods used within the area of complex function theory to:
  • determine function properties; continuity and analyticity complex functions
  • apply Cauchy-Riemann equation to functions to determine if a function is analytical
  • Möbius transform and its special cases, including dilation, translation, rotation, and inversion.
  • Cauchy’s integral theorem and integral formula
  • design of Möbius transform based on mapping points
  • curve integrals, closed curve integrals, finding critical points for functions
  • apply Cauchy’s integral theorem og formula to analytical functions
• apply the presented concepts, theories and methods used within the areas of series theory and Fourier transformation to:
  •  
  • Convergence tests Series and sequences
  • 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 functions
  • development 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 for Fourier Transforms.  
  • Power Series, coefficients and the center
  • Radius of Convergence R – Cauchy-Hadamards formular
  • Taylor and Maclaurin power series
  • Fourier Series
  • Fourier Integrals
  • Amplitude and phase specters by the Fourier transform
  • Apply the presented concepts, theories and methods used within the area of vector analysis to:
  • Parametriske beskrivelser af kurser og flader
  • Konservative felter og solenoidale felter
  • Begrebet potentialefunktion
  • Anvendelse af Jacobianten i forbindelse med variabelsubstitutioner
  • Fremstille parametriske repræsentationer af kurver og flader ud fra verbale, formelle eller grafiske beskrivelser
  • Skitsere givne kurser og flader
  • Evaluere kurveintegraler, dobbeltintegraler, fladeintegraler og volumenintegraler
  • Foretage variabelskift under anvendelse af Jacobiant
  • 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 givent konservativt felt samt kontrollere løsningen
  • Evaluere vejuafhængige kurveintegraler ved at finde stamfunktion
  • Parameterization of curve integrals and path-independent curve integrals

Kompetencer

• The Fourier transform
• Amplitude and phase specters by the Fourier transform
• Fremstille parametriske repræsentationer af kurver og flader ud fra verbale, formelle eller grafiske beskrivelser

Undervisningsform

Forelæsninger, opgaveregning, workshops, selvstudie.

Eksamen

Prøver