# Introductory course in Laplace transforms and complex analysis 2022

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!

## Syllabus

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.

## Timeplan

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

 Monday 8/08 Tuesday 9/08 Wednesday 10/08 Thursday 11/08 Friday 12/08 Lecture Lecture&Examples Lecture&Examples Lecture&Examples Lecture/Review Lecture Lecture&Examples Lecture&Examples Lecture&Examples Lecture/Review Exercises Lecture&Examples Lecture&Examples Exercise session Exercise session No lecture! (Matriculation Ceremony) Lecture (Laplace) Lecture (Laplace) No lecture (We start at 14:15) Lecture (Laplace) Lecture (Laplace) Lecture (Laplace) Lecture (Laplace) Exercise session Exercise session Exercise session Exercise session Exercise session

## Lecture plan

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).

 Topic Notes and Exercises Recordings Complex numbers and operations on complex numbers. The geometric interpretation of complex numbers. Polar form of complex numbers. Multiplication and division in polar form. Triangle inequalities. Integer powers of a complex number and De Moivre’s formula. Exponential form of a complex number. Roots. Exercises with solutions. A couple of links that might be useful\interesting: Euler's identity complex exponentiation notes_complex_numbers.pdf solved_exercises_1.pdf Rec1 Rec2 Rec3 Relation between sinusoidal signals and complex numbers. AC circuits. Series RLC circuit. Example. Remarks on circles, disks and sets in the complex plane. example_ac_circuits.pdf remarks_sets_etc.pdf Rec1 Rec2 Rec3 Introduction to complex functions with examples. Limit in the complex plane. Continuity and derivative of complex functions. Analytic functions. Exercises with solutions. notes_complex_analysis_1.pdf solved_exercises_2.pdf Rec1 Rec2 Rec3 Cauchy-Riemann equations. Laplace's equation. Harmonic functions. Exercise session on the topics of the first two lectures (last hour). In the notes, N.C. stands for "necessary condition", while S.C. for "sufficient condition" notes_complex_analysis_2.pdf exercises_1.pdf exercises_1_solutions.pdf Rec1 Rec2 Complex exponential, trigonometric, hyperbolic, and logarithmic functions with examples. Hints on general powers. Remarks on partial fraction decomposition. Exercise session about the topics of the last three lectures on complex analysis (last hour). notes_complex_analysis_3.pdf exercises_2.pdf exercises_2_solutions.pdf partial_fraction_dec_ex_sol.pdf Rec1 Rec2 Rec3 Introduction to the Laplace transform (definition and some results about existence and uniqueness). First examples of Laplace transforms. s-Sfifting theorem. The recordings are mute (I forgot to turn on the microphone!) Exercise session in the last hour. notes_laplace_transf_1.pdf exercises_laplace_1.pdf exercises_laplace_1_sol.pdf Rec1 Rec2 Transforms of first and second derivatives. Solution of Initial Value Problems with the Laplace transform method. Examples. Recording of the second hour missing. Exercise session in the last hour. notes_laplace_transf_2.pdf exercises_laplace_2.pdf exercises_laplace_2_sol.pdf Rec1 Transform of integral. Unit step function and second shifting theorem. Applications to IVPs with examples. Exercise session in the last hour. notes_laplace_transf_3.pdf exercises_laplace_3.pdf exercises_laplace_3_sol.pdf Rec1 Dirac's delta function and applications to IVPs. Convolution. Hints on differentiation and integration of transforms. Examples. Notes and exercises included in the previous lecture Rec1 Rec2