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Fourier Analysis, 6 ECTS credits.
/Fourieranalys/
For:
IT
Mat
Y
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Prel. scheduled
hours: 62
Rec. self-study hours: 98
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Area of Education: Science
Main field of studies: Mathematics, Applied Mathematics, Electrical engineering, Applied Physics,Biomedical Engineering
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Advancement level
(G1, G2, A): G2
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Aim:
The course covers Fourier series as well as Fourier, Laplace and z-transforms in a unified treatment based on the foundations of distribution theory and complex analysis. It will give mathematical knowledge fundamental for treatment of problems in system engineering and physics. It is also a preparation for courses in partial differential equations. After a completed course, the student will be able to:
- Differentiate, integrate and transform distributions in one variable with particular emphasis on the Dirac distribution and its derivatives.
- Calculate Fourier series for simple periodic functions and distributions and determine convergence properties and estimate approximation errors in the mean.
- Solve linear differential equations with constant coefficients using distributions and Fourier- and Laplace transforms and linear difference equations using z-transforms.
- Using the complex inversion integral, in combination with residue calculus, to calculate inverse Laplace and z-transforms.
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
Calculus (one and several variables), Linear Algebra and Complex analysis or equivalent.
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
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Organisation:
Lectures, problem classes.
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Course contents:
Basic distribution theory in one variable. Basic properties of Fourier series, Fourier, Laplace and z-transforms. Convergence of Fourier series, point wise and in the mean. Parseval's formula. Integrals with a parameter. The Fourier transform. The inversion formula. Rules of manipulation. The convolution formula. Parseval's formula. Inversion formulas and their validity. Convolutions and their transforms. Transforms of distributions. Applications to engineering and science.
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Course literature:
Lecture notes with exercises available on the course home page. Mathematical tables "Formelsamling för Fourieranalys".
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Examination: |
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Written examination
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6 ECTS
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Course language is Swedish.
Department offering the course: MAI.
Director of Studies: Jesper Thorén
Examiner: Mats Aigner
Link to the course homepage at the department
Course Syllabus in Swedish
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