# Advanced Analysis

Structure Type: | Study unit |
---|---|

Code: | IZS9109 |

Curriculum: | IT 2020 |

Level: | Bachelor of Engineering |

Year of Study: | 3 (2022-2023) |

Semester: | Autumn |

Credits: | 4 cr |

Responsible Teacher: | Mäkelä, Jarmo |

Language of Instruction: | English |

## Learning Outcomes

During the Advanced Analysis course the student will deepen the knowledge gained during the courses of Differential calculus and Analysis. The largest individual part of the course consists of complex analysis. In that part of the course the student will learn to differentiate and integrate functions of a complex variable. Complex analysis is largely based on power series, and therefore, as a preparation for the complex analysis, the student is first familiarized with the power series of functions of a real variable, their divergence and convergence. Complex analysis, in the level considered in this course, is intented to provide the student the necessary knowledge for the Integral Transforms course. Power series may also be used for the solution of differential equations.

The student is also made familiar with basic partial differential equations and their solution by means of the separation of variables. The course end with a brief review on the calculus of variations, which may be viewed as a one step forward from the usual differential- and integral calculus. Among other things, classical mechanics as a whole may be derived from a single principle formulated by means of the calculus of variations.

## Student's Workload

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

## Prerequisites / Recommended Optional Courses

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

## Contents

1. A brief summary of the basic properties of series and sequences.

2. Power series of functions of a real variable.

3. Convergence and divergence of power series, radius of convergence.

4. Functions of a complex variable.

5. Analytic functions, Cauchy-Riemann equations.

6. Line integrals of functions of a complex variable.

7. Cauchy’s integral theorem.

8. Power series of analytic functions; radius of convergence.

9. Pole of a function of a complex variable.

10. Laurent series.

11. Residue.

12. The residue theorem.

13. Integration of functions of a real variable by means of the residue theorem.

14. Solution of linear differential equations by means of the power series method.

15. Solution of partial differential equations by means of the separation of variables.

16. Elements of the calculus of variations.

17. Constrained variational problems.

18. Lagrange’s formulation of classical mechanics: The principle of least action.

## Recommended or Required Reading and Other Learning Resources/Tools

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

## Mode of Delivery / Planned Learning Activities and Teaching Methods

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

## Assessment Criteria

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.

## Assessment Methods

Homework exercises and an examination.