Introductory course in Laplace transforms and complex analysis 2022

NB: the page for 2023 is here: https://wiki.math.ntnu.no/tma4120/2023h/forkurs/start

Lecturer: Andrea Leone

This course provides a brief introduction to Laplace transforms and complex analysis for the incoming two-year master's students who don't have the necessary backgrounds or wish a repetition of the subjects. The students do not need any prior knowledge other than that provided by undergraduate courses in mathematics (basic linear algebra and mathematical analysis). The course is voluntary and gives no credits. There is no exam and no registration.

Lectures are planned with physical attendance and will not be in a hybrid format or streamed online. Since the rooms are equipped with cameras, I will order automatic recording of the lectures and upload them on Panopto. Some of you may be able to attend the course only during one of the two weeks, hence we will deal with complex analysis during week 32 and Laplace transforms during week 33 instead of starting with both the topics from the beginning.

The course is held in English!

The course is based on Chapter 6 (Laplace Transforms) and Chapter 13 (Complex Numbers and Functions) from:
Erwin Kreyszig
Advanced Engineering Mathematics
10th Edition
2011
ISBN 978-0-470-64613-7

• Intro complex analysis : We start with a review of complex numbers (algebraic, polar and exponential form) and algebraic operations with complex numbers. We go on with complex functions and define analytic functions. We introduce Cauchy-Riemann Equations and show how they are related to Laplace equation. We conclude with some examples of complex functions.

• Intro Laplace transforms: We define the Laplace transform and show some properties. We proceed with Laplace transforms of derivatives and integrals and show how to solve ODEs by transforming them into algebraic problems. We study the Laplace transform of the unit step function and of the Dirac's delta function and show some applications. Finally, we state the convolution theorem and, if time allows, we mention differentiation and integration of transforms.

There is no need to buy the book. I will upload notes on this site before (hopefully!) each lecture.

The course will be held in August 2022, in week 32 (August 8th – 12th) and in week 33 (August 16th – 19th).

You can check this link to visualize a learning flow map with the logical relations among the topics of the course. This way you could understand what is your level of knowledge (starting from the node "complex numbers", try to see if you understand the various relations and remember the various topics). More resources on self-assessment material will follow probably (to access the flow map you have to register to the faceit site, which is used, among the others, by professors at the IE faculty of NTNU. No personal data is collected about you).