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# Integral Transforms

Rakennetyyppi: Opintojakso IZS9110 IT 2020 Insinööri (AMK) 3 (2022-2023) 3 op Mäkelä, Jarmo Englanti

## Osaamistavoitteet

The solution of various mathematical problems becomes considerably easier, if the functions under study are replaced by their integral transforms. When a function is replaced by its integral transform, the function transforms, by means of integration, to a new function of a new variable. The most imnportant integral transforms are the Fourier transform, which is applied, in particular, in the analysis of vtyhe oscillatory phenomena, and the Laplace transform, which is used in the solution of differential equations. Closely related to the differential equations are the so-called difference equations, which have sequences as their solutions. The difference equations may be solved by means of the so-called z-transforms. In this course, which is heavily based on the complex analysis learned in the Advanced Analysis course, the student learns the basics of all these transforms and their application.

## Opiskelijan työmäärä

The total amount of student's work is 108 h, which contains 56 h of contact studies. The assessment of student’s own learning 1 h is included in contact lessons.

## Edeltävät opinnot / Suositellut valinnaiset opinnot

Analysis: Differential- and Integral Calculus basics and Differential equations and series.

## Sisältö

1. A brief summary of complex analysis and the thorem of residues from the Advanced Analysis course.
2. Fourier series; the Dirichlet theorem.
3. Complex Fourier series.
4. A analysis of standing waves by means of the Fourier series.
5. Evaluation of selected series by means of the Fourier series.
6. Solution of differential equations by means of the Fourier series.
7. Fourier transforms and inverse transforms.
8. Examples of the determination of the Fourier transforms and inverse transforms by means of the therem of residues.
9. Solution of differential equations by means of the Fourier transforms.
10. Laplace transform.
11. Inverse Laplace transforms.
12. Bromwich integral. (Or Mellin’s inverse formula)
13. Determination of inverse Laplace transforms by means of residues.
14. Solution of linear differential equations by means of the Laplace transforms.
15. Solution of systems of differential equations using Laplace transforms.
16. The convolution theorem.
17. Shift of the origin in the Laplace transforms and inverse transforms.
18. Causal sequence.
19. z-transform of a causal sequence.
20. Inverse z-transform.
21. Determination of inverse z-transforms by means of residues.
22. Difference equations.
23. Solution of difference equations by means of z-trasforms.
24. Elements of Mellin transforms.

## Opiskelumateriaali

Kreyszig, E: "Advanced Engineering Mathematics", John Wiley & Sons; the material prepared by the lecturer.

## Opetusmuoto / Opetusmenetelmät

Theory, examples and exercises during the lectures. Homework exercises.

## Arviointikriteerit

Grade 1: The student knows those subjects of the course, which are necessary for the forthcoming studies and working life.
Grade 3: The student is well-abled to utilize the course contents.
Grade 5: The student is able to apply creatively the contents of the course.

## Arviointimenetelmät

Homework exercises and an examination.